Testing the Combination of the Anti-cancer Drugs XL184 (Cabozantinib) and Nivolumab in Patients With Advanced Cancer and HIV

Inclusion Criteria:

* Age \>= 18 years. Children are excluded from this study, but will be eligible for future pediatric trials
* For the six-patient safety cohort, subjects must have histologically or cytologically confirmed advanced solid tumors that are metastatic or recurrent, and require palliative systemic treatment, for which there are either Food and Drug Administration (FDA) approved indications for XL184 (cabozantinib) or nivolumab or have at least phase 2 data clearly indicating activity (such as renal cell carcinoma \[RCC\], hepatocellular carcinoma \[HCC\], medullary thyroid carcinoma \[MTC\], melanoma, non-small cell lung cancer \[NSCLC\], head and neck cancer, urothelial carcinoma, small cell lung cancer \[SCLC\], radioiodine-refractory differentiated thyroid cancer, ovarian cancer, castration-resistant prostate carcinoma \[CRPC\], and triple-negative breast cancer \[TNBC\]). Subjects must have progressed, or are intolerant, or decline systemic therapy associated with clinically significant survival benefit if checkpoint blockade is not an approved or accepted treatment. The expansion cohort is limited to subjects with KS. Histologic, cytologic, and pathologic confirmation of KS is required
* Any number of prior cancer therapies will be permitted, including treatment naive subjects. (Note: For KS, treatment naive asymptomatic subjects will be permitted. But treatment naive KS subjects with visceral symptomatic disease or complicated KS HHV 8 disease including Castleman's disease will be excluded and should receive front-line standard of care)
* Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 (Karnofsky \>= 80%)
* Subjects with tumors other than KS must have evaluable disease
* Absolute neutrophil count \>= 1,000/mcL
* Platelets \>= 75,000/mcL
* Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (If, however, the participant has Gilbert's disease or unconjugated hyperbilirubinemia that is considered to be secondary to antiretroviral therapy, then the total bilirubin must be =\< 3 x ULN)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN
* Creatinine =\< 1.5 institutional ULN OR creatinine clearance (CrCl) \>= 50 mL/min (if using the Cockcroft-Gault formula)
* Hemoglobin \>= 9 g/dL
* CD4 count \>= 50/mcL
* Subjects must have known HIV infection as below: Serologic documentation of HIV infection at any time prior to study entry, as evidenced by positive enzyme-linked immunosorbent assay (ELISA), positive Western blot, or any other federally approved licensed HIV test. Alternatively, this documentation may include a record that another physician has documented that the participant has HIV infection based on prior ELISA and Western blot, or other approved diagnostic tests. Subjects must receive appropriate care and treatment for HIV infection. An eligible patient should be on anti-retroviral therapy (ART) that is not strongly CYP3A4 inhibiting or otherwise prohibited by the protocol (e.g. drug-drug interactions) or the patient must be converted to one of these regimens before starting investigational therapy in order to avoid dose modulation of cabozantinib
* Life expectancy of \>= 12 weeks
* For subjects with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated
* Subjects with a history of hepatitis C virus (HCV) infection must have been treated and cured including self-cured cases. For subjects with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load
* Subjects with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
* The effects of nivolumab and XL184 (cabozantinib) on the developing human fetus are unknown. For this reason, women of child-bearing potential (WOCBP) and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. WOCBP should use an adequate method to avoid pregnancy for 5 months after the last dose of investigational drug. WOCBP must have a negative serum or urine pregnancy test within 72 hours prior to the start of receiving the first dose of the study medication. Men who are sexually active with WOCBP must use any contraceptive method with a failure rate of less than 1% per year. Men receiving nivolumab and who are sexually active with WOCBP will be instructed to adhere to contraception for a period of 7 months after the last dose of investigational product. Women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile as well as azoospermic men) do not require contraception

  * Women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. In addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL
* Ability to understand and the willingness to sign a written informed consent document. Subjects with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible

Exclusion Criteria:

* For the safety run-in cohort, subjects who have received prior XL184 (cabozantinib), PD-1/PD-L1 inhibitor, or VEGFR inhibitor are ineligible. Prior treatment with these agents is allowed for the expansion KS cohort
* Subjects on potent CYP3A4-inhibiting agents are ineligible, such as:

  * Antiretroviral: ritonavir, cobicistat, indinavir, atazanavir, delavirdine
  * Antibiotics: clarithromycin, erythromycin, telithromycin, troleandomycin
  * Antifungals: itraconazole, ketoconazole, voriconazole, fluconazole, posaconazole
  * Antidepressants: nefazodone
  * Antidiuretic: conivaptan
  * Gastrointestinal (GI): cimetidine, aprepitant
  * Hepatitis C: boceprevir, telaprevir
  * Miscellaneous: Seville oranges, grapefruit, or grapefruit juice and/or pummelos, star fruit, exotic citrus fruits, or grapefruit hybrids Of note, to meet the eligibility requirement, subjects are allowed to convert their antiretroviral medications to one of the regimens not including potent CYP3A4-inhibiting agents, when the subjects have progressed, are intolerant, or decline the standard systemic therapy for their advanced tumors.

Subjects must receive appropriate care and treatment for HIV infection, including antiretroviral medications, when clinically indicated (including no ART) and should be under the care of a physician experienced in HIV management. Subjects will be eligible provided there is no intention to initiate therapy or the regimen has been stable for at least 4 weeks with no intention to change the regimen within 8 weeks following study entry.

To enroll in the study, the participants should be on the protocol accepted ART as long as they are receiving XL184 (cabozantinib)

* Subjects who have had cytotoxic chemotherapy (including investigational cytotoxic chemotherapy) or biologic agents (e.g., cytokines or antibodies) within 3 weeks, or nitrosoureas/mitomycin C within 6 weeks before the first dose of study treatment, or those who have not recovered from adverse events (AEs) due to agents administered more than 4 weeks earlier
* The subject has received radiation therapy:

  * To the thoracic cavity, abdomen, or pelvis within 4 weeks before the first dose of study treatment, or has ongoing complications, or is without complete recovery and healing from prior radiation therapy
  * To bone or brain metastases within 14 days before the first dose of study treatment
  * To any other site(s) within 21 days before the first dose of study treatment
* Subjects who are receiving any other investigational agents
* Subjects must be either off corticosteroids, or on a stable or decreasing dose of =\< 10 mg daily prednisone (or equivalent) for at least 2 weeks prior to enrollment. Inhaled or topical steroids are permitted in the absence of active autoimmune disease
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to nivolumab or XL184 (cabozantinib)
* The subject has prothrombin time (PT)/ international normalized ratio (INR) or partial thromboplastin time (PTT) test \>= 1.3 x the laboratory ULN within 7 days before the first dose of study treatment
* The subject has a primary brain tumor, active brain metastases or epidural disease. Subjects with brain metastases previously treated with whole brain radiation or radiosurgery or participants with epidural disease previously treated with radiation or surgery who are asymptomatic and do not require steroid treatment for at least 2 weeks before starting study treatment are eligible. Subjects with treated brain metastasis should not take enzyme-inducing anticonvulsive therapies (EIACDs) within 2 weeks of registration, though non-enzyme inducing anticonvulsive drugs such as levetiracetam are allowed. Neurosurgical resection of brain metastases or brain biopsy is permitted if completed at least 3 months before starting study treatment. Baseline brain imaging with contrast-enhanced CT or MRI scans for participants with known brain metastases is required to confirm eligibility. Subjects with untreated central nervous system (CNS) metastases are eligible if they are not symptomatic and the lesions are less than 1 cm in size. CNS metastases should be stable for at least 4 weeks, neurologically asymptomatic and without corticosteroid treatment at time of first dose of study treatment
* Concomitant anticoagulation with coumarin agents (e.g., warfarin), direct thrombin inhibitors (e.g., dabigatran), direct factor Xa inhibitor betrixaban, or platelet inhibitors (e.g., clopidogrel). Allowed anticoagulants are the following:

  * Prophylactic use of low-dose aspirin for cardio-protection (per local applicable guidelines) and low-dose low molecular weight heparins (LMWH)
  * Therapeutic doses of LMWH or anticoagulation with direct factor Xa inhibitors rivaroxaban, edoxaban, or apixaban in subjects without known brain metastases who are on a stable dose of the anticoagulant for at least 1 week before first dose of study treatment without clinically significant hemorrhagic complications from the anticoagulation regimen or the tumor
* The subject has experienced any of the following:

  * Clinically-significant gastrointestinal bleeding within 6 months before the first dose of study treatment
  * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months before the first dose of study treatment
  * Any other signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment
* The subject has radiographic evidence of cavitating pulmonary lesion(s)
* The subject has tumor in contact with, invading, or encasing any major blood vessels
* The subject has evidence of tumor invading the GI tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor within 28 days before the first treatment
* The subject has uncontrolled and significant cardiovascular disorders:

  * Congestive heart failure (CHF): New York Heart Association (NYHA) class III (moderate) or class IV (severe) at the time of screening
  * Concurrent uncontrolled hypertension defined as sustained blood pressure (BP) \> 140 mm Hg systolic, or \> 90 mm Hg diastolic despite optimal antihypertensive treatment within 7 days of the first dose of study treatment
  * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 500 ms within 28 days before treatment

    * Note: If initial QTcF is found to be \> 500 ms, two additional electrocardiograms (EKGs) separated by at least 3 minutes should be performed. If the average of these three consecutive results for QTcF is =\< 500 ms, the subject meets eligibility in this regard
  * Any of the following within 6 months before the first dose of study treatment:

    * Unstable angina pectoris
    * Clinically-significant cardiac arrhythmias
    * Stroke
    * Myocardial infarction
    * Subjects with a venous filter (e.g. vena cava filter) are not eligible
    * Thromboembolic event
* The subject has uncontrolled and significant disorders particularly those associated with a high risk of perforation or fistula formation including:

  * Any of the following within 28 days before the first dose of study treatment:

    * Active and symptomatic peptic ulcer disease
    * Evidence of active or acute diverticulitis, intra-abdominal abscess, GI obstruction and abdominal carcinomatosis which are known risk factors for bowel perforation should be evaluated for the potential need for additional treatment before coming on study
  * Any of following within 6 months before the first dose of study treatment:

    * Abdominal fistula
    * Gastrointestinal perforation
    * Bowel obstruction or gastric outlet obstruction intra-abdominal abscess. Note: Complete resolution of an intra-abdominal abscess must be confirmed prior to initiating treatment with cabozantinib even if the abscess occurred more than 6 months before
  * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement within 3 months before the first dose of study therapy
* Subjects with active autoimmune disease or history of autoimmune disease that might recur, which may affect vital organ function or require immune suppressive treatment including systemic corticosteroids, should be excluded. These include but are not limited to subjects with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as systemic lupus erythematosus (SLE), connective tissue diseases, scleroderma, inflammatory bowel disease (IBD), Crohn's, ulcerative colitis, hepatitis; and subjects with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome should be excluded because of the risk of recurrence or exacerbation of disease. Subjects with vitiligo, endocrine deficiencies including thyroiditis managed with replacement hormones including physiologic corticosteroids are eligible. Subjects with rheumatoid arthritis and other arthropathies, Sjogren's syndrome and psoriasis controlled with topical medication and subjects with positive serology, such as antinuclear antibodies (ANA), anti-thyroid antibodies should be evaluated for the presence of target organ involvement and potential need for systemic treatment but should otherwise be eligible. Subjects are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger (precipitating event)
* Major surgery (e.g., laparoscopic nephrectomy, GI surgery, removal or biopsy of brain metastasis) within 2 weeks before first dose of study treatment. Minor surgeries within 10 days before first dose of study treatment. Subjects must have complete wound healing from major surgery or minor surgery before first dose of study treatment. Subjects with clinically relevant ongoing complications from prior surgery are not eligible
* The subject is unable to swallow tablets
* History of organ transplant or stem cell transplant
* Subjects with uncontrolled intercurrent illness
* Subjects with psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study because XL184 (cabozantinib) has the potential for teratogenic or abortifacient effects, and the effects of nivolumab on the developing fetus are not well known. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother, breastfeeding must be discontinued if the mother is treated with XL184 (cabozantinib) or nivolumab