Sirolimus, Docetaxel, and Carboplatin in Treating Patients With Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria:

* Signed informed consent form (ICF) providing agreement to adhere to the dosing schedule, report for all trial visits and authorization, use and release of health and research trial information
* Histologically or cytologically confirmed carcinoma of the prostate (excluding neuroendocrine differentiation or squamous cell histology)
* Ongoing gonadal androgen deprivation therapy with gonadotropin-releasing hormone (GnRH) analogues, antagonists or orchiectomy; patients who have not had an orchiectomy must be maintained on effective GnRH analogue/antagonist therapy
* Castration resistant prostate cancer as defined by serum testosterone \< 50 ng/ml and at least one of the following:

  * PSA level of at least 2 ng/ml that has risen on at least 2 successive occasions at least 1 week apart
  * Evaluable disease progression by modified RECIST 1.1
  * Progression of metastatic bone disease on bone scan with \> 2 new lesions
* Prior therapy with abiraterone, enzalutamide and/or docetaxel; if a patient has not received docetaxel or cabazitaxel chemotherapy, the patient must be informed of this treatment choice as an alternative; if the patient has received docetaxel or cabazitaxel chemotherapy or refuses one of both of these therapies, this rationale must be documented and the patient is then eligible; patient must be offered and made aware of all Food and Drug Administration (FDA)-approved treatment options; patients with bone only disease may not have received radium-223
* The presence of metastatic disease amenable to computed tomography (CT) or ultrasound guided biopsy; this may include thoracolumbar vertebral bodies, pelvis, femur or humerus or soft tissue or nodal metastasis amenable to biopsy (excluding lung or pleural lesions)
* Agree to participate in biopsy of metastatic lesion during the study at day 21
* Eastern Cooperative Oncology Group (ECOG) performance status of =\< 1
* Life expectancy \>= 12 weeks
* No prior malignancy is allowed except:

  * Adequately treated basal cell or squamous cell skin cancer or
  * In situ carcinoma of any site or
  * Other adequately treated malignancy for which the patient is currently disease free for at least one year
* Patients with any prior chemotherapy regimens are eligible
* Patients with disease only in the bone may not have received Xofigo/radium 223 to avoid ongoing DNA damage in bone marrow
* Patients who are or are not receiving bisphosphonates or denosumab are eligible; bisphosphonates or denosumab should not be initiated after registration and during active treatment
* Absolute neutrophil count \>= 1.5 x 10\^9 cells/L (within 14 days prior to registration)
* Hemoglobin (Hgb) \>= 9.0 g/dL (within 14 days prior to registration)
* Platelets \>= 100,000 x 10\^9/L (within 14 days prior to registration)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 1.5 x upper limit of normal (ULN) (within 14 days prior to registration)
* Total bilirubin =\< 1.5 x ULN (within 14 days prior to registration)
* Serum creatinine \< 1.5 X institutional ULN mg/dL OR estimated glomerular filtration rate (eGFR) \>= 50 mL/min (within 14 days prior to registration)

Exclusion Criteria:

* Patients currently receiving active therapy for other neoplastic disorders
* Patients with histologic evidence of small cell carcinoma of the prostate will not be eligible
* Patients with disease only in the bone previously treated with radium-223 will not be eligible
* Known parenchymal brain metastasis
* Active or symptomatic viral hepatitis or chronic liver disease
* Estimated creatinine clearance less than 50 ml/minute
* Clinically significant heart disease as evidenced by myocardial infarction, or arterial thrombotic events in the past 6 months, severe or unstable angina, or New York Heart Association (NYHA) class II-IV heart disease
* Atrial fibrillation, or other cardiac arrhythmia requiring medical therapy
* Administration of an investigational therapeutic within 30 days of cycle 1, day -2
* Patients with dementia/psychiatric illness/social situations that would limit compliance with study requirements or would prohibit the understanding and/or giving of informed consent
* Patients with medical conditions, which, in the opinion of the investigators, would jeopardize either the patient or the integrity of the data obtained will not be eligible
* Any condition which, in the opinion of the investigator, would preclude participation in this trial
* Patients on anticoagulation therapy which cannot be held for metastatic biopsies