Study of Olaparib Maintenance Following Cabazitaxel-Carbo in Men With AVPC

The goal of this clinical research study is to learn if olaparib, when given after treatment with cabazitaxel, carboplatin, and prednisone, can help to control aggressive variant prostate cancer (AVPC). The safety of these drugs will also be studied. This is an investigational study. Cabazitaxel and carboplatin are FDA approved and commercially available for the treatment of certain types of prostate cancer. Prednisone is FDA-approved and commercially available as a corticosteroid. Olaparib is FDA approved and commercially available for the treatment of certain types of ovarian cancer. The combination of cabazitaxel and carboplatin followed by olaparib in this study is investigational. The study doctor can describe how the study drugs are designed to work. Up to 96 participants will be enrolled on this study. All will take part at MD Anderson.

Inclusion Criteria: 1. Provision of informed consent prior to any study specific procedures. 2. Provision of informed consent for genetic research. If a patient declines to participate in the genetic research, there will be no penalty or loss of benefit to the patient. The patient will not be excluded from other aspects of the study described in this Clinical Study Protocol, so long as they consent to that part. 3. Patients must agree to tissue collection for correlative studies at the specified timepoints. If patient has undergone a recent tissue collection without intervening treatment since, that can be retrieved and is deemed of sufficient quantity by the PI to undertake the proposed correlative studies, it may be used as the baseline. 4. Male aged 18 years and above. 5. Histologically or cytologically confirmed prostate carcinoma. 6. Presence of metastatic disease documented on imaging studies (bone scan, computed tomography (CT) and/or magnetic resonance imaging (MRI) scans). 7. Patients must meet at least one of the following AVPC criteria: i. Histologically proven small cell (neuroendocrine) prostate carcinoma ii. Exclusive visceral metastases. iii. Predominantly lytic bone metastases identified by plain x-ray or CT scan. iv. Bulky (\>/= 5cm in longest dimension) lymphadenopathy or high-grade tumor mass in prostate/pelvis. v. Low PSA (\/= 20) bone metastases. vi. Elevated serum lactate dehydrogenase (\>/=2 x upper limit of normal) or elevated serumcarcinoembryonic antigen (\>/= 2 x upper limit of normal ) in the absence of other etiologies. vii. Short interval (\/= 1.0 ng/mL; b) New or increasing non-bone disease (RECIST); c) Positive bone scan with 2 or more new lesions (PCWG3). d) Increasing symptoms unequivocally attributed to disease progression as judged by the treating physician and the PI. 10. Surgically or ongoing medically castrated, with baseline testosterone levels of \/= 1.5 x 10\^9/L (unless due to bone marrow infiltration by tumor, in which case ANC \>1,000/mm3 is allowed) iii. White blood cells (WBC) \>3x10\^9/L (unless due to bone marrow infiltration by tumor, in which case WBC \>2x109/L is allowed) iv. No features suggestive of myelodysplastic syndrome/acute myeloid leukemia on peripheral blood smear v. Platelet count \>/= 100 x 10\^9/L (unless due to bone marrow infiltration by tumor, in which case platelet \>/=50,000/ mm3 is allowed) vi. Total bilirubin \