Durvalumab and Tremelimumab in Treating Chemotherapy Naive Patients With Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: * Written informed consent. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Life expectancy of \>= 52 weeks. * Hemoglobin \>= 11.0 g/dL. * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L (\> 1500 per mm\^3). * Platelet count \>= 100 x 10\^9/L (\>100,000 per mm\^3). * Serum bilirubin =\< 1.5 x institutional upper limit of normal (ULN). This will not apply to subjects with confirmed Gilbert's syndrome (persistent or recurrent hyperbilirubinemia that is predominantly unconjugated in the absence of hemolysis or hepatic pathology), who will be allowed only in consultation with their physician. * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]/alanine aminotransferase \[ALT\] serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal. * Serum creatinine clearance (CL) \> 40 mL/min by the Cockcroft-Gault formula (Cockcroft and Gault 1976) or by 24-hour urine collection for determination of creatinine clearance. * Subject is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up. * Consent to MD Anderson laboratory protocol PA13-0291 and LAB02-152. * Willing to have peripheral blood mononuclear cells and bone marrow biopsies to be collected prior to receiving the first dose of durvalumab and tremelimumab, after 2-doses and 4-doses of durvalumab and tremelimu…