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 * Pl…