Elesclomol Sodium and Paclitaxel in Treating Patients With Recurrent or Persistent Ovarian Epithelial Cancer, Fallopian Tube Cancer, or Primary Peritoneal Cancer

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 Response Evaluation Criteria in Solid Tumors (RECIST) 1.1; measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded); each lesion must be \>= 10 mm when measured by computed tomography (CT), magnetic resonance imaging (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 Gynecologic Oncology Group (GOG) protocol, if one exists; in general, this would refer to any active GOG phase III protocol for the same patient population * Patients must have a GOG performance status of 0, 1, or 2 * Patients must have baseline lactate dehydrogenase (LDH) levels =\< 0.8 x upper limit of normal (ULN) * Recovery from effects of recent surgery, radiotherapy, or chemotherapy * Patients should be free of active infection requiring antibiotics (with the exception of uncomplicated urinary tract infection \[UTI\]) * Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration * Any other prior therapy directed at the malignant tumor, including biological and immunologic agents, must be discontinued at least three weeks prior to registration * Prior therapy * 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 intraperitoneal therapy, high-dose therapy, consolidation, non-cytotoxic agents or extended therapy administered after surgical or non-surgical assessment * Patients must have NOT received any additional cytotoxic chemotherapy for management of recurrent or persistent disease, including retreatment with initial chemotherapy regimens; (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) hormones, monoclonal antibodies, cytokines, and small-molecule inhibitors of signal transduction * Patients must be considered platinum resistant or refractory according to standard GOG criteria, i.e., have had a treatment-free interval following platinum of less than 6 months, or have progressed during platinum-based therapy * Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl * Platelets greater than or equal to 100,000/mcl * Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN) * Bilirubin less than or equal to 1.5 x ULN * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) less than or equal to 3 x ULN * Alkaline phosphatase less than or equal to 2.5 x ULN * Neurologic function: neuropathy (sensory and motor) less than or equal to Common Terminology Criteria for Adverse Events (CTCAE) 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 study entry and be practicing an effective form of contraception * Cautions and prohibited medications/treatments * Since elesclomol is a substrate for cytochrome P450 family 2, subfamily C, polypeptide 9 (CYP2C9), cytochrome P450 family 2, subfamily D, polypeptide 6 (CYP2D6), cytochrome P450 family 2, subfamily C, polypeptide 19 (CYP2C19), and cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) and an inducer of CYP3A4, cytochrome P450 family 1, subfamily A, polypeptide 2 (CYP1A2), cytochrome P450 family 2, subfamily A, polypeptide 6 (CYP2A6), and cytochrome P450 family 2, subfamily E, polypeptide 1 (CY2E1), it is recommended that the following be used with caution: sensitive substrates of CYP3A4, CYP1A2, and CY2E1, and strong inhibitors and inducers of CYP2C9, CYP2D6, CYP2C19, and CYP3A4 Exclusion Criteria: * Patients who have had prior therapy with elesclomol or prior second-line cytotoxic chemotherapy * 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 are excluded if there is any evidence of other malignancy being present within the last three 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, fallopian tube, or primary peritoneal 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, fallopian tube, or primary peritoneal 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 who are pregnant or breastfeeding