Belinostat and Carboplatin in Treating Patients With Recurrent or Persistent Ovarian Epithelial Cancer, Fallopian Tube Cancer, or Primary Peritoneal Cancer That Did Not Respond to Carboplatin or Cisplatin

Inclusion Criteria: * Patients must have recurrent or persistent epithelial ovarian, fallopian tube, or primary peritoneal carcinoma; histologic documentation of the original primary tumor is required via the pathology report * Patients with the following histologic epithelial cell types are eligible: Serous adenocarcinoma, endometrioid adenocarcinoma, mucinous adenocarcinoma, undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's Tumor, or adenocarcinoma not otherwise specified (N.O.S.) * All patients must have measurable disease as defined by RECIST 1.1; measureable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest dimension to be recorded); each lesion must be \>= 10 mm when measured by CT, MRI or caliper measurement by clinical exam; or \>= 20 mm when measured by chest x-ray; lymph nodes must be \>= 15 mm in short axis when measured by CT or MRI * Patients must have at least one "target lesion" to be used to assess response on this protocol as defined by RECIST 1.1; tumors within a previously irradiated field will be designated as "non-target" lesions unless progression is documented or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy * Patients must not be eligible for a higher priority GOG protocol, if one exists; in general, this would refer to any active GOG Phase III or Rare Tumor protocol for the same patient population * Patients must have a GOG Performance Status of 0, 1, or 2 * Recovery from effects of recent surgery, radiotherapy, or chemotherapy * Patients should be free of active infection requiring antibiotics (with the exception of uncomplicated UTI) * Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration; continuation of hormone replacement therapy is permitted * Any other prior therapy directed at the malignant tumor, including biological and immunologic agents, must be discontinued at least three weeks prior to registration * Patients must have had one prior platinum-based chemotherapeutic regimen for management of primary disease containing carboplatin, cisplatin, or another organoplatinum compound; this initial treatment may have included non-cytotoxic therapy, intraperitoneal therapy, high-dose therapy, consolidation, or extended therapy administered after surgical or non-surgical assessment * Patients must be considered platinum resistant or refractory according to the following criteria: * Patients must have had progression of disease on or within 6 months of their last platinum dose * Progression of disease is defined according to RECIST 1.1 * The development of CA125 elevation on or within 6 months of last platinum treatment, in the absence of radiographic progression according to RECIST 1.1, is not considered platinum resistance for the purposes of this study * Patients who have NOT received prior therapy with taxane-based chemotherapy MUST receive a second regimen that includes paclitaxel or docetaxel * Patients must be considered paclitaxel-resistant, i.e., have had a treatment-free interval following paclitaxel of less than six months, or have progressed during paclitaxel-based therapy * Patients must have NOT received any additional cytotoxic chemotherapy for management of recurrent or persistent disease, including retreatment with initial chemotherapy regimens except as noted above; (note: Optimal evaluation of the safety and efficacy of new chemotherapy regimens is best performed in patients with minimal prior therapy; non-investigational therapy, such as retreatment with platinum and/or paclitaxel, is non-curative in the setting of recurrent disease, and can generally be safely administered to patients following participation in a phase II trial) * Patients are allowed to receive, but are not required to receive, one additional non-cytotoxic regimen for management of recurrent or persistent disease according to the following definition: * Non-cytotoxic (biologic or cytostatic) agents include (but are not limited to) monoclonal antibodies, cytokines, and small-molecule inhibitors of signal transduction * Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl, equivalent to the NCI Common Terminology Criteria (CTCAE v3.0) grade 1 * Platelets greater than or equal to 100,000/mcl * Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN), per CTCAE v.3.0 grade 1 * Bilirubin less than or equal to 1.5 x ULN (CTCAE v.3.0 grade 1) * SGOT (AST) less than or equal to 3 x ULN (per the CTCAE v.3.0 grade 1) * Alkaline phosphatase less than or equal to 2.5 x ULN (CTCAE v.3.0 grade 1) * Neuropathy (sensory and motor) less than or equal to the CTCAE v3.0 grade 1 * Patients must have signed an approved informed consent and authorization permitting release of personal health information * Patients must meet pre-entry requirements * Patients of childbearing potential must have a negative serum pregnancy test prior to the study entry and be practicing an effective form of contraception Exclusion Criteria: * Patients who have had prior therapy with Belinostat (PXD101) or other HDAC inhibitors * Patients who have received radiation to more than 25% of marrow-bearing areas * Patients with a history of other invasive malignancies, with the exception of non-melanoma skin cancer, and other specific malignancies as noted below, are excluded if there is any evidence of other malignancy being present within the last five years; patients are also excluded if their previous cancer treatment contraindicates this protocol therapy * Patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis OTHER THAN for the treatment of ovarian cancer are excluded; prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease * Patients who have received prior chemotherapy for any abdominal or pelvic tumor OTHER THAN for the treatment of ovarian cancer are excluded; patients may have received prior adjuvant chemotherapy for localized breast cancer, provided that it was completed more than three years prior to registration, and that the patient remains free of recurrent or metastatic disease * Patients must not use concomitant medications on PXD infusion days that may cause Torsade de Pointes; medications associated with this risk include: * Amiodarone, Arsenic trioxide, Bepridil, Cisapride, Disopyramide, Dofetilide, Droperidol, Erythromycin, Felbamate, Flecainide, Fluoxetine, Halofantrine, Haloperidol, Ibutilide, Levofloxacin, Mesoridazine, Pentamidine, Procainamide, Quinidine, Sotalol, Sparfloxacin, or Thioridazine * Patients with significant cardiovascular disease defined as: * Unstable angina pectoris, uncontrolled hypertension (blood pressure \> 150/90 despite maximal medical therapy), congestive heart failure related to primary cardiac disease, any condition requiring anti-arrhythmic therapy, ischemic or severe valvular heart disease, or history of myocardial infarction within 6 months of trial entry * Patients who are pregnant or nursing