Pembrolizumab in Metastatic Castration Resistant Prostate Cancer (mCRPC) With or Without DNA Damage Repair Defects

Inclusion Criteria: 1. Documented histology of adenocarcinoma of the prostate. 2. Metastatic castration resistant prostate cancer with castrate-level testosterone (\<50 ng/dL). a. Subjects must maintain a castrate-level testosterone during the study. 3. Disease progression defined by one or more of the following three criteria: 1. Prostate-specific antigen (PSA) \> 2.0 ng/mL and rising PSA by at least 2 consecutive measurements a minimum of 1-week apart. 2. Soft tissue progression as defined by RECIST v1.1 criteria. 3. Bone disease progression as defined by Prostate Cancer Clinical Trials Working Group 3 (PCWG3). 4. Have received prior secondary hormonal therapy including abiraterone, enzalutamide and/or apalutamide. 5. Be taking prednisone at a dose of ≤ 10mg/day, 7 days prior to starting treatment (Cycle 1, Day 1). 6. Be willing and able to provide written informed consent/assent for the trial. 7. Be \>= 18 years of age on day of signing informed consent. 8. Patients must agree to have a tumor tissue biopsy at baseline, and there must be a lesion that can be biopsied with acceptable clinical risk as judged by the investigator. 1. Patients with inconclusive DNA damage repair status testing on this baseline biopsy must have one of the following (per the investigator's discretion): (i) Sufficient archival tissue, or (ii) An additional biopsy attempt. 2. Patients with previously identified homozygous deletion or deleterious germline or somatic mutation(s) in DNA damage repair gene(s) (such as BReast CAncer gene 1 (BRCA1) , BReast CAncer gene 2 (BRCA2), and ATM) identified in a Clinical Laboratory Improvement Amendments of 1988 (CLIA)-certified laboratory are allowed in Group 2. (i) Somatic mutation(s) in DNA damage repair gene(s) needs to be identified on the biopsy of a castration-resistant tumor site (ii) Archival FFPE tissue will be requested for determination of MSI (if not already assessed by gene sequencing) signature status.(iia) A formalin-fixed paraffin-embedded (FFPE) tumor specimen in a paraffin block (preferred) or at least 10 slides containing unstained, freshly cut, serial sections must be available along with an associated pathology report before study enrollment. (iii) If archival FFPE tissue is unable to be obtained or is insufficient, patients will be required to undergo tumor tissue biopsy if feasible for determination of MSI signature status. 3. Patients with germline mutation(s) in mismatch repair (MMR) gene(s) (i.e. Lynch syndrome),or have previously identified Microsatellite instability (MSI)-high tumor by Polymerase chain reaction (PCR) or MMR deficient tumor by Immunohistochemistry (IHC) are also allowed in Group 2. (i) Archival FFPE tissue will be requested for determination of Fanconi anaemia (FA)/BRCA signature status. (ia) A formalin-fixed paraffin-embedded (FFPE) tumor specimen in a paraffin block (preferred) or at least 10 slides containing unstained, freshly cut, serial sections must be available along with an associated pathology report before study enrollment. (ii) If archival FFPE tissue is unable to be obtained or is insufficient, patients will be required to undergo tumor tissue biopsy if feasible for determination of FA/BRCA signature status. 9. If group 1 is not filled, patients may proceed onto treatment without the completion of tests for DNA repair status. Once group 1 is filled, patients cannot be enrolled onto the study or start treatment until DNA damage repair status is successfully determined for study group placement. a. Patients will be replaced if they have tissues that are not evaluable for DNA repair mutations 10. Patients must be willing to provide archival tissue from prior biopsy or surgery for prostate cancer, if available. a. A formalin-fixed paraffin-embedded (FFPE) tumor specimen in a paraffin block (preferred) or at least 10 slides containing unstained, freshly cut, serial sections must be available along with an associated pathology report before study enrollment. 11. Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale. 12. Patients must discontinue antiandrogen therapy (i.e. bicalutamide, flutamide, nilutamide) at least 4-6 weeks prior to registration with no evidence of PSA decline after washout. 1. Bicalutamide: Washout period at least 6 weeks 2. Flutamide and nilutamide: Washout period at least 4 weeks 13. Patients must discontinue therapies for mCRPC, with the exception of Gonadotropin-releasing hormone (GnRH) agent, for 14 days, with the exception of anti-androgens with which there may be a withdrawal PSA response. 1. Prior chemotherapy is allowed if no progression of disease on chemotherapy. 2. Prior treatment with sipuleucel-T, radium-223, or poly ADP-ribose polymerase (PARP) inhibitor (e.g. olaparib) is allowed. 3. Tissue biopsy may be performed during washout period. 14. Demonstrate adequate organ function as defined as follows, all screening labs should be performed within 28 days of treatment initiation. 1. Hematological: (i) Absolute neutrophil count (ANC): \>= 1,500 /microliters (mcL) (ii) Platelets: \>= 100,000 / mcL (iii) Hemoglobin: \>= 9 g/dL or \>= 5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment) 2. Renal: (i) Serum creatinine OR Measured or calculated a creatinine clearance (GFR can also be used in place of creatinine or CrCl): \<=1.5 X upper limit of normal (ULN) OR \>= 60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN. Creatinine clearance should be calculated per institutional standard 3. Hepatic: (i) Serum total bilirubin: \<= 1.5 X ULN (ii) Aspartate Aminotransferase (AST) (SGOT) and Alanine Aminotransferase (ALT) (SGPT): \<= 2.5 X ULN OR \<= 5 X ULN for subjects with liver metastases (iii) Albumin: \>= 2.5 mg/dL 4. Coagulation: (i) International Normalized Ratio (INR) or Prothrombin Time (PT): \<= 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or Partial Thromboplastin Time (PTT) is within therapeutic range of intended use of anticoagulants (ii) Activated Partial Thromboplastin Time (aPTT): \<= 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants 15. Male subjects of childbearing potential must agree to use an adequate method of contraception, starting with the first dose of study therapy through 120 days after the last dose of study therapy. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject. Exclusion Criteria: 1. Significant liver metastasis. 2. Prior taxane-based chemotherapy with progressive disease on chemotherapy. 1. Prior docetaxel for metastatic hormone sensitive prostate cancer is allowed, if no progression of disease on docetaxel as defined by RECIST v1.1 and PCWG3. 2. Prior taxane-based chemotherapy (i.e. docetaxel or cabazitaxel with or without platinum agent) for mCRPC is allowed if no progression of disease on chemotherapy as defined by RECIST v1.1 and PCWG3. 3. Is currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment. 4. Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy \>10mg/day or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. 5. Has a known history of active Bacillus Tuberculosis (TB). 6. Hypersensitivity to pembrolizumab or any of its excipients. 7. Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., \<= Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. 8. Has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e., \<= Grade 1 or at baseline) from adverse events due to a previously administered agent. 1. Note: Subjects with \<= Grade 2 neuropathy are an exception to this criterion and may qualify for the study. 2. Note: If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. 9. Has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer. 10. Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. 11. Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy \<= 10 mg of prednisone/day for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. 12. Has known history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or current pneumonitis/interstitial lung disease. 13. Has an active infection requiring systemic therapy. 14. Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. 15. Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. 16. Is expecting to father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. 17. Has received prior therapy with an anti-programmed cell death protein 1 (PD-1), anti-PD-L1, or anti-PD-L2 agent. 18. Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). 19. Has known active Hepatitis B (HBV) (i.e Hepatitis B surface antigen (HBsAg) reactive) or Hepatitis C (e.g., hepatitis C virus (HCV) RNA \[qualitative\] is detected). 20. Has received a live vaccine or live-attenuated vaccine within 30 days prior to the first dose of study drug. Administration of killed vaccines is allowed.