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

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

What this study is about