Testing the Addition of an Immunotherapy Drug, Tremelimumab, to the PARP Inhibition Drug, Olaparib, for Recurrent Ovarian, Fallopian Tube or Peritoneal Cancer

Inclusion Criteria: * Patients must have platinum-sensitive, recurrent high-grade serous or high-grade endometrioid (grade 3) ovarian, primary peritoneal, or fallopian tube cancer. Patients with other histologies are also eligible, provided that the patient has a known deleterious germline or somatic BRCA1 or BRCA2 mutation identified through testing at a clinical laboratory. Submission of BRCA testing results (germline and/or somatic) is required for all patients. * Platinum-sensitive disease defined as no clinical or radiographic evidence of disease recurrence for \> 6 months (or 182 days) after last receipt of platinum-based therapy. The date should be calculated from the last administered dose of platinum therapy. * Patients must have had response (complete or partial) to their prior line of platinum therapy and cannot have had progression through prior platinum-based therapy. * Patients must have RECIST 1.1 measurable disease. Patients with biochemical recurrence based on CA125 levels alone are not eligible. * Prior therapy: * Prior chemotherapy must have included a first-line platinum-based regimen with or without consolidation chemotherapy. * Prior bevacizumab therapy as a component of frontline or recurrent treatment is permitted. * Patients may have received an unlimited number of platinum-based therapies in the recurrent setting. * Patients may have received up to one non-platinum-based line of therapy in the recurrent setting. Prior hormonal therapy will …