A Study of CHeckpoint Inhibitors in Men With prOgressive Metastatic Castrate Resistant Prostate Cancer Characterized by a Mismatch Repair Deficiency or Biallelic CDK12 Inactivation

Inclusion Criteria:

* Subject must be 18 years of age or older at the time the Informed Consent is signed.
* The subject (or legally acceptable representative if applicable) must provide written informed consent for the trial.
* Pathologic diagnosis of prostate cancer of adenocarcinoma or small cell histology.
* Metastatic disease as documented by technetium-99m (99mTc) bone scan or metastatic lesions by computed tomography (CT) or magnetic resonance imaging (MRI) scans (visceral or lymph node disease). CT-portion of FDG-PET/CT or scan may be used for eligibility. NaF PET-CT is an alternative to 99mTc bone scan. If lymph node metastasis is the only evidence of metastatic disease, it must be 1.5 cm in short axis and above the level of the iliac bifurcation. Imaging studies for the purpose of determining eligibility must be completed within 60 days of Day 1.
* Progressive castration resistant prostate cancer as defined by serum testosterone \< 50 ng/mL and one of the following:

  * PSA progression confirmed per Prostate Cancer Clinical Trials Working Group (PCWG3),
  * Radiographic progression of soft tissues according to Response Evaluation Criteria in Solid Tumors, version 1.1 (iRECIST 1.1) modified based on PCWG3, or radiographic progression of bone according to PCWG3.
* Prior use of a novel AR signaling inhibitor for 4 weeks, including abiraterone acetate plus prednisone/prednisolone, enzalutamide, apalutamide, and/or darolutamide.

NOTE: These AR signaling inhibitors may have been used for mCSPC, M0CRPC, and/or mCRPC.

* Ongoing surgical or medical castration, with testosterone levels of \<50 ng/dL. If the subject is being treated with GnRH analogs (subject who has not undergone bilateral orchiectomy), this therapy must have been initiated at least 30 weeks prior to initiation of pembrolizumab and must be continued throughout the study.
* ECOG PS grade of 0-1.
* Metastatic lesion that is amenable to biopsy and performed within 180 days of Day 1.
* dMMR or CDK12-/- as determined by somatic tumor DNA NGS.

  * Either monoallelic or biallelic inactivation of CDK12 on NGS is considered sufficient for eligibility purposes.
  * MMR genes include: MLH1, MSH2, MLH3, PMS1, MSH6, and PMS2.
  * dMMR is established by MSI-H on NGS. However, for "weak" MMR genes, inclusive of PMS2 and MSH6, monoallelic inactivation will be allowed for eligibility purposes if and only if there is at least MSI-low or hypermutation that is concomitantly present.
  * If there is biallelic inactivation of "strong" MMR genes (MLH1 and MSH2), then patients must manifest MSI-H. However, if the tumor DNA utilized for MSI analysis was obtained \> 6 months prior to NGS, then the NGS should be repeated to determine if MSI-H has developed. Monoallelic inactivation of "strong" MMR genes will be allowed if MSI-H is present; in this scenario, it is presumed that biallelic inactivation is present but the second inactivating event was not detected due to technical issues such as low sensitivity for copy loss.
* Adequate organ function:

  * Hemoglobin (hgb) \> 9.0 g/dL,
  * Absolute neutrophil count (ANC) \> 1500/ uL,
  * Platelets \> 100,000/ uL,
  * Total bilirubin 1.5 x ULN OR direct bilirubin ULN for participants with total bilirubin levels \>1.5 x ULN
  * ALT and AST 2.5 x ULN ( 5 x ULN for participants with liver metastases) (Child-Pugh class A and B allowed; Child-Pugh class C is excluded).
  * Creatinine \< (2.0 mg/dL) during screening evaluation (\>2.0 is allowed if EGFR \>30 mL/min/1.73 m2).
* Subject must agree to use contraception during the treatment period plus an additional 120 days after the last dose of study treatment and must refrain from donating sperm during this period.

Exclusion Criteria:

* Brain metastases.
* Prior treatment with an anti-PD1, anti-PDL1, or anti PD L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, CTLA-4, OX 40, CD137).
* Anti-neoplastic therapies for prostate cancer must be completed \> 2 weeks prior to Day 1 (initiation of pembrolizumab); chemotherapy must be completed \> 4 weeks prior to Day 1.
* Has received prior systemic anti-cancer therapy including investigational agents within 4 weeks \[could consider shorter interval for kinase inhibitors or other short half-life drugs\] prior to \[randomization /allocation\].

Note: Participants must have recovered from all AEs due to previous therapies to Grade 1 or baseline. Participants with Grade 2 neuropathy may be eligible.

* Herbal and non-herbal products that may decrease PSA levels other than medical castration and megestrol (up to 40 mg/day is allowed) for hot flashes.
* Has received prior radiotherapy within 2 weeks of start of study treatment. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation ( 2 weeks of radiotherapy) to non-CNS disease.
* Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to the first dose of study treatment.

Note: Participants who have entered the follow-up phase of an investigational study may participate as long as it has been 4 weeks after the last dose of the previous investigational agent.

* If a subject has undergone major surgery, they must have recovered adequately from the toxicities or complications from the intervention within 4 weeks prior to starting therapy.
* History of non-prostate active malignancy requiring treatment in the 24 months prior to Day 1 except for non-muscle invasive urothelial cancer and non-melanoma skin cancer.
* Active infection or conditions requiring treatment with antibiotics.
* Immunosuppressive doses of systemic medications, such as corticosteroids (doses \> 10 mg/day prednisone or equivalent), within 2 weeks of Day 1.
* Has a history of (non-infectious) pneumonitis that required steroids or has current pneumonitis.
* Has a known history of active TB (Bacillus Tuberculosis).
* Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the subject's participation for the full duration of the study, or is not in the best interest of the subject to participate, in the opinion of the treating investigator.
* Active autoimmune disease or a documented history of autoimmune disease that requires immunosuppressive medications within the last two years (e.g., chronic steroids, methotrexate, tacrolimus, etc.).
* Active or chronic hepatitis B or hepatitis C disease as determined by hepatitis B surface antigen (HBsAg), hepatitis B core antibody, or hepatitis C antibody (anti-HCV) positivity at screening. If positive, further testing of quantitative levels to rule out active infection is required.
* History of positive test for human immunodeficiency virus (HIV). NOTE: Hepatitis B and C and HIV testing is NOT required during screening.
* Vaccinated with a live vaccine within 30 days of enrollment.
* Has severe hypersensitivity ( Grade 3) to pembrolizumab and/or any of its excipients.
* Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial.
* Subject is planning to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment.