Tremelimumab + Durvalumab(MEDI4736)+ Belinostat in Urothelial Carcinoma

Inclusion Criteria:

* Male or female subject aged \>= 18 years
* Histologically confirmed urothelial carcinoma with metastatic disease or with unresectable, locally advanced disease. Variant histology, including, but not limited to, neuroendocrine, sarcomatoid, and squamous differentiation are permitted to enroll
* Patients must meet one or more of the following criteria:

  * Has progressed on at least one prior therapy; or
  * Has declined standard therapy; or
  * Is not a suitable candidate for standard therapy
  * The discussion regarding the choice of standard therapy offered, if available, and patient's choice and reason(s) to decline standard therapy should be documented clearly in the research chart.
  * Patients may have progressed on immune checkpoint inhibitor therapy
* Body weight \> 30 kg
* Subject must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria by computed tomography (CT) or magnetic resonance imaging (MRI)
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 1
* Absolute neutrophil count (ANC) \>= 1500/mm\^3
* Platelet count \>= 100,000/mm\^3
* Hemoglobin \>= 10 g/dL Hemoglobin \< 10 g/dL due to hematuria at the time of treatment is acceptable as long as there is evidence of adequate bone marrow function in the opinion of the investigator.
* Total Bilirubin =\< 1.5 x institutional upper limit of normal (ULN)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN

  * Patients with liver metastases will be allowed to enroll with AST and ALT levels =\< 5 x ULN
* Estimated creatinine clearance \>= 30 mL/min by Cockcroft-Gault formula
* Evidence of post-menopausal status or negative urinary or serum pregnancy test for female pre-menopausal patients. Women will be considered post-menopausal if they have been amenorrheic for 12 months without an alternative medical cause. The following age-specific requirements apply:

  * Women \< 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of exogenous hormonal treatments and if they have luteinizing hormone and follicle-stimulating hormone levels in the post-menopausal range for the institution or underwent surgical sterilization (bilateral oophorectomy or hysterectomy)
  * Women \>= 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments, had radiation-induced menopause with last menses \> 1 year ago, had chemotherapy-induced menopause with last menses \> 1 year ago, or underwent surgical sterilization (bilateral oophorectomy, bilateral salpingectomy or hysterectomy)
* Highly effective contraception for both male and female subjects throughout the study and at least 4 months after last study treatment administration
* Recovery to baseline or =\< grade 1 Common Terminology Criteria for Adverse Events (CTCAE) version (v)5.0 from toxicities related to any prior treatments, unless adverse event (AE)(s) are clinically non-significant and/or stable on supportive therapy
* Able to provide informed consent and willing to sign an approved consent form that conforms to federal and institutional guidelines
* Estimated life expectancy of at least 12 weeks

Exclusion Criteria:

* Homozygous for UGT1A1\*28 allele or heterozygous or homozygous for UGT1A1\*60 orGilbert syndrome
* Subject has received systemic antineoplastic therapy (including unconjugated therapeutic antibodies and toxin immunoconjugates) or any investigational therapy =\< 14 days or within 5 half-lives before starting study treatment, whichever is shorter
* Prior treatment with durvalumab
* Subject has received radiotherapy =\< 14 days before the first dose of study treatment. Localized radiation therapy for the treatment of symptomatic bone metastasis is allowed during that timeframe
* Subjects who have undergone major surgery =\< 3 weeks before starting study drug or who have not fully recovered from major surgery
* Known additional malignancy that is active and/or progressive and requiring treatment.
* Known brain metastases or cranial epidural disease

  * Note: Brain metastases or cranial epidural disease adequately treated with radiotherapy and/or surgery and stable for at least 4 weeks before the first dose of study treatment will be allowed on trial. Subjects must be neurologically asymptomatic and without corticosteroid treatment at the time of the first dose of study treatment
* Current evidence of uncontrolled, clinically significant intercurrent illness including, but not limited to, the following conditions:

  * Cardiovascular disorders:

    * Congestive heart failure New York Heart Association Class 3 or 4, unstable angina pectoris, serious cardiac arrhythmias
    * Stroke (including transient ischemic attack \[TIA\]), myocardial infarction (MI), or other ischemic events, or thromboembolic event (e.g., deep venous thrombosis, pulmonary embolism) within 3 months before the first dose
    * Uncontrolled hypertension defined as a sustained systolic blood pressure \>= 160 mmHg or a diastolic blood pressure \>= 100 mmHg despite optimal management

      * Note: Patients with uncontrolled hypertension who are not optimally managed may be rescreened once controlled hypertension is achieved
      * Note: Patients with uncorrectable prolonged corrected QT (QTc) (Bezet formula) \> 480 msec or concomitant use of medications(s) with a known risk of inducing Torsade de Pointes if such treatment cannot be discontinued or switched to a different medication before starting the study drug
      * Note: If a single electrocardiogram (ECG) shows a QTc with an absolute value \> 480 msec, two additional ECGs approximately 2 minutes apart must be performed within 30 minutes of the initial ECG, and the average of these three consecutive results for QTc will be used
  * Adrenal insufficiency
  * Interstitial lung disease (ILD)
* Subjects with congenital long QT syndrome
* Patients currently on or who will require valproic acid for any medical condition during the study
* Active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease \[e.g., colitis or Crohn's disease\], diverticulitis \[except for diverticulosis\], systemic lupus erythematosus, Sarcoidosis syndrome, or Wegener syndrome \[granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis, etc.\]). The following are exceptions to this criterion:

  * Patients with vitiligo or alopecia
  * Patients with hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone replacement
  * Any chronic skin condition that does not require systemic therapy
  * Patients without active disease in the last 5 years may be included but only after consultation with the principal investigator
  * Patients with celiac disease controlled by diet alone
* Prior randomization or treatment in a previous durvalumab clinical study regardless of treatment arm assignment.
* Patients who have received prior anti -PD-1, anti PD-L1 or anti CTLA-4:

  * Must not have experienced a toxicity that led to permanent discontinuation of prior immunotherapy.
  * All AEs while receiving prior immunotherapy must have completely resolved or resolved to baseline prior to screening for this study.
  * Must not have experienced a ≥Grade 3 immune related AE or an immune related neurologic or ocular AE of any grade while receiving prior immunotherapy. NOTE: Patients with endocrine AE of ≤Grade 2 are permitted to enroll if they are stably maintained on appropriate replacement therapy and are asymptomatic.
  * Must not have required the use of additional immunosuppression other than corticosteroids for the management of an AE, not have experienced recurrence of an AE if re-challenged, and not currently require maintenance doses of \> 10 mg prednisone or equivalent per day.
* Current or prior use of immunosuppressive medication within 14 days of cycle one day one, EXCEPT for the following permitted steroids:

  * Intranasal, inhaled, topical steroids, eye drops or local steroid injection (e.g., intra-articular injection);
  * Systemic corticosteroids at physiologic doses =\< 10 mg/day of prednisone or equivalent;
  * Steroids as premedication for hypersensitivity reactions (e.g., computed tomography (CT) scan premedication)
* History of active primary immunodeficiency
* Known human immunodeficiency virus (HIV) infection with a detectable viral load at the time of screening

  * Note: Patients on effective antiretroviral therapy with an undetectable viral load at the time of screening are eligible for this trial
* Active infection including tuberculosis (clinical evaluation that includes clinical history, physical examination, and radiographic findings, and tuberculosis (TB) testing in line with local practice), hepatitis B (known positive hepatitis B virus \[HBV\] surface antigen (HBsAg) result), or hepatitis C

  * Note: Patients with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody \[anti-HBc\] and absence of HBsAg) are eligible. Patients positive for hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative for HCV ribonucleic acid (RNA)
* Live attenuated vaccinations within 4 weeks of cycle one day one and while on trial is prohibited
* Known prior severe hypersensitivity to investigational product or any component in its formulations, including known severe hypersensitivity reactions to monoclonal antibodies (National Cancer Institute \[NCI\] CTCAE version \[v\]5.0 grade \>= 3)
* Subjects taking prohibited medications. A washout period of prohibited medications for a period of at least 5 half-lives or as clinically indicated should occur before the start of treatment
* Any other condition or disease that would, in the Investigator's judgment, contraindicate the subject's participation in the clinical study due to safety concerns or compliance with clinical study procedures. (e.g. extensive peritoneal carcinomatosis).
* Homozygous for UGT1A1\*28 allele or heterozygous or homozygous for UGT1A1\* 60 or Gilbert syndrome.