PRINCE (PSMA-lutetium Radionuclide Therapy and ImmuNotherapy in Prostate CancEr)

Inclusion Criteria: Patients must meet the following criteria for study entry: 1. Patient who are at least 18 years of age who have provided written informed consent. 2. Histologically confirmed adenocarcinoma of the prostate without neuroendocrine or small cell differentiation. 3. Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 1 (see Appendix 1). 4. Patients must have progressed on prior enzalutamide, abiraterone and/or apalutamide for treatment of prostate cancer. 5. Determination of disease progression on second generation androgen receptor targeted agent determined by the local investigator. Progressive disease is defined by PCWG3 as any one of the following: * PSA progression: minimum of two rising PSA values from a baseline measurement with an interval of ≥ 1 week between each measurement. The PSA value at screening should be ≥ 1ng/ml. * Soft tissue or visceral disease progression as per modified RECIST 1.1 criteria (see Appendix 2) * Bone progression: ≥ 2 new lesions on bone scan (Appendix 2) 6. At least 2 weeks since the completion of surgery or radiotherapy prior to registration. Any clinically relevant sequelae from the surgery or radiotherapy must have improved to grade 1 prior to registration. 7. Prior surgical orchiectomy or chemical castration maintained on luteinizing hormone-releasing hormone (LHRH) analog (agonist or antagonist). Patients without prior surgical castration must be currently taking and willing to continue luteinizing hormone-releasing hormone (LHRH) analog (agonist or antagonist) therapy throughout the duration of study treatment. 8. Serum testosterone levels ≤ 50ng/dL. (≤ 1.75nmol/L) within 28 days before registration. 9. Imaging evidence of metastatic disease documented with either bone scan or CT scan (Appendix 2). 10. Prior prostate cancer vaccine therapy, radiation therapy, systemic therapies, diethylstilboestrol (DES) or other estrogens are allowed up to 28 days prior to trial registration. Note: bicalutamide flutamide or nilutamide must be discontinued within 4 weeks of registration. 11. Significant PSMA avidity on 68Ga/18F-PSMA PET/CT, defined as a minimum uptake of SUVmax 20 at a site of disease, and SUVmax \> 10 at other sites of disease ≥10mm (unless subject to factors explaining a lower uptake, e.g. respiratory motion, reconstruction artefact). 12. Patients must have a life expectancy ≥ 24 weeks. 13. Patients must agree to use a highly effective form of contraception for the entire duration of the study plus an additional 120 days (a spermatogenesis cycle) after the last dose of study treatment and refrain from donating sperm during this period (see section 10.3.3). 14. Patients must be willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled assessments. 15. Patients must have adequate bone marrow, hepatic and renal function documented within 28 days of registration, defined as: * Haemoglobin ≥90 g/L independent of transfusions (no red blood cell transfusion in last 4 weeks) * White blood cells \>3x109/L * Absolute neutrophil count ≥1.5x109/L * Platelets ≥100 x109/L * Total bilirubin ≤1.5 x upper limit of normal (ULN) except for patients with known Gilbert's syndrome. * Aspartate transaminase (AST) (SGOT) and alanine transaminase (ALT) (SGPT) ≤2.5 × ULN or ≤5 × ULN for participants with liver metastases. * Serum creatinine ≤1.5 x ULN or a calculated creatinine clearance \> 50mL/min (Chronic Kidney Disease Epidemiology (CKD-EPI) equation for patients with creatinine levels above institutional normal. * Albumin \>30 g/dl * International normalized ratio (INR), prothrombin time (PT), activated partial thromboplastin time (aPTT) ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants. 16. Patients who are deemed by PSMA imaging to have readily accessible disease will be required to consent to 3 serial tumour biopsies - at screening, post combination treatment (at any time between weeks 2-4) and on progression Exclusion Criteria: 1. Site(s) of disease that are FDG positive with low PSMA expression defined by PSMA SUVmax \< 10. 2. Previous history or presence of brain metastases or leptomeningeal metastases. 3. Any prior exposure to anti-PD-1, anti-PD-L1/L2, anti-CTLA-4 antibody or any other antibody or drug specifically targeting T cell co-stimulation or checkpoint pathway. 4. Any prior treatment with cabazitaxel. 5. Any prior exposure to 177Lu-PSMA. 6. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. 7. Patients with active, known or suspected autoimmune disease including Sjogren's syndrome. Patients with vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are eligible. 8. Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients. 9. Participants must have recovered from all AEs due to previous therapies to ≤Grade 1 or baseline. Participants with ≤ Grade 2 neuropathy may be eligible. 10. Patients with a condition requiring systemic treatment with either corticosteroids (\> 10mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration. Inhaled or topical steroids, and adrenal replacement doses ≤ 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. 11. Other malignancies within the previous 2-years other than, melanoma in situ, basal cell or squamous cell carcinomas of skin with a \> 30% probability of recurrence within 12 months. 12. Positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection. 13. Patient has a known history of Human Immunodeficiency Virus (HIV). 14. Patients with symptomatic or impending cord compression unless appropriately treated beforehand and clinically stable for ≥ 4 weeks. 15. Previous history of interstitial lung disease or non-infectious pneumonitis. 16. Recent administration of a live vaccine within 30 days prior to the first dose of study drug. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guérin (BCG), and typhoid vaccine. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (eg, FluMist®) are live attenuated vaccines and are not allowed. 17. Recent administration of the influenza vaccine (within 30 days of registration).