Belinostat in Treating Patients With Advanced Ovarian Epithelial Cancer, Primary Peritoneal Cancer, or Fallopian Tube Cancer or Ovarian Low Malignant Potential Tumors

Inclusion Criteria: * Histologically/cytologically confirmed ovarian epithelial cancer, primary peritoneal carcinoma or fallopian tube cancer that recurred despite initial platinumbased therapy OR micropapillary/borderline (Low Malignant Potential) tumors of ovary * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan * Patients in the platinum resistant (progression within 6 months of platinum based therapy) group: must have had no more than a total of 3 prior chemotherapy regimens; at least one prior regimen will have contained a platinum agent (carboplatin or cisplatin) * Patients with micropapillary or borderline (LMP) tumors: must have had no more than a total of 3 prior chemotherapy regimens * Patients must have completed any prior chemotherapy, radiotherapy or surgery at least 4 weeks (at least 6 weeks for nitrosureas and mitomycin C) before study entry and patients must have recovered from the toxic effects from any prior therapy; patients must not have had more than 40% of their bone marrow radiated and must have either measurable disease outside the field or progression post radiation therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Leukocytes \>= 3.0 x 10\^9/L * Absolute neutrophil count \>= 1.5 x 10\^9/L * Platelets \>= 100 x 10\^9/L * Total bilirubin =\< institutional upper limit of normal * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\])/ alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal * Creatinine =\< institutional upper limit of normal OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Biopsies are not mandatory but patients will be asked to undergo tumor biopsy once before and once during experimental therapy if considered medically safe for them to do so; patients must be willing to have the peripheral blood mononuclear cell (PBMC) procured prior to and during the treatment * Eligibility of patients receiving any medications or substances known to affect or with the potential to affect the activity or pharmacokinetics of PXD101 will be determined following review of their case by the Principal Investigator; efforts should be made to switch patients who are taking enzyme-inducing anticonvulsant agents to other medications * Women of child-bearing potential and their partners must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * A marked baseline prolongation of QT/corrected QT (QTc) interval, e.g., repeated demonstration of a QTc interval \> 500msec; long QT syndrome; the required use of concomitant medication on PXD101 infusion days that may cause Torsade de Pointes (disopyramide, dofetilide, ibutilide, procainamide, quinidine, sotalol, bepridil, amiodarone, arsenic trioxide, cisapride, lidoflazine, clarithromycin, erythromycin, halofantrine, pentamidine, sparfloxacin, domperidone, droperidol, chlorpromazine, haloperidol, mesoridazine, thioridazine, pimozide \& methadone) * Significant cardiovascular disease including unstable angina pectoris, uncontrolled hypertension, congestive heart failure related to primary cardiac disease, a condition requiring anti-arrythmic therapy, ischemic or severe valvular heart disease, or a myocardial infarction within 6 months prior to the trial entry * Patients may not be receiving any other investigational agents * Patients with known brain metastases should be excluded from this clinical trial * History of allergic reactions attributed to sulfonamides, arginine and compounds of similar chemical or biologic composition to PXD101 * Patients should not have taken valproic acid, another histone deacytelase inhibitor, for at least 2 weeks prior to enrollment * 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 * Patients with bowel obstruction would not be eligible because of compromised functional status unless they are on parenteral support * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with PXD101 * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible