A Phase 1b Study of Paclitaxel And Ricolinostat For The Treatment Of Gynecological Cancer

Inclusion Criteria: * Participants must have recurrent or persistent epithelial ovarian, fallopian tube or primary peritoneal carcinoma, recurrent endometrial cancer, or recurrent cervical cancer. Histologic documentation of the original primary tumor is required via the pathology report. * Participants must have measurable disease by RECIST 1.1 criteria. See Section 11 for the evaluation of measurable disease. * Participants must have had at least one prior platinum-based chemotherapeutic regimen for management of primary disease (e.g., a regimen containing carboplatin, cisplatin, or another organoplatinum compound). This initial treatment may have included intraperitoneal therapy, consolidation, biologic/targeted (non-cytotoxic) agents (e.g., bevacizumab) or extended therapy administered after surgical or non-surgical assessment. Participants are allowed to receive, but are not required to receive, biologic/targeted (non-cytotoxic) therapy as part of their primary treatment regimen. * Participants must have recurrence within 12 months of their last platinum-containing regimen. * Age 18 years or older * ECOG performance status 0 or 1 * Life expectancy of greater than 16 weeks * Participants must have normal organ and marrow function as defined below: * Leukocytes ≥3,000/mcL * Absolute neutrophil count ≥1,500/mcL * Platelets ≥100,000/mcL * Total bilirubin within normal institutional limits * AST(SGOT)/ALT(SGPT) ≤2.5 × institutional upper limit of normal * Creatinine within normal institutional limits OR * Creatinine clearance ≥60 mL/min/1.73 m2 for participants with creatinine levels above institutional normal. * Previous toxicities from previous treatment must have resolved to grade 1 or less * For patients in expansion cohort B, stable Grade 2 neuropathy will be allowed. * The effects of both paclitaxel and oral ricolinostat on the developing human fetus are unknown. For this reason, women of child-bearing potential and men 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 she or her partner is participating in this study, she should inform her treating physician immediately. * Participants must be able and willing to swallow pills and to absorb oral medications. * Ability to understand and the willingness to sign a written informed consent document * Participants must be able and willing to follow protocol instructions and schedules. Exclusion Criteria: * Participants 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. In addition, no small molecule kinase inhibitors or any other type of investigational agent may have been administered within 4 weeks before first dose of study treatment. * Participants may not be receiving any other investigational agents for treatment of their cancer. * No hormonal therapy is allowed within 1 week of initiating study treatment. * Participants may not have had radiation to \>25% of the bone marrow. * Prior treatment with a histone deacetylase inhibitor. * Prior treatment with weekly paclitaxel for recurrent or persistent disease is not allowed. Participants may have received weekly paclitaxel as part of treatment for newly diagnosed cancer, but may not have received it as maintenance therapy following their initial therapy with platinum and taxane therapy. * Participants with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * History of severe allergic reactions attributed to compounds of similar chemical or biologic composition to either paclitaxel or Ricolinostat. Patients who require administration of paclitaxel through a desensitization procedure are not eligible for this study. * 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 chronic viral illnesses such as HIV-positivity and active hepatitis B or C are ineligible because they are at increased risk of lethal infections when treated with marrow-suppressive therapy. * Any signs, symptoms, and/or radiographic evidence of a complete or partial bowel obstruction * 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 three years. Patients are also excluded if their previous cancer treatment contraindicates this protocol therapy. * Carcinoma in situ of the breast or cervix * Primary endometrial cancer meeting the following conditions: Stage not greater than IA, grade 1 or 2, no more than superficial myometrial invasion, without vascular or lymphatic invasion; no poorly differentiated subtypes, including papillary serous, clear cell, or other FIGO grade 3 lesions. * 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 within the last three years 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 the patient remains free of recurrent or metastatic disease. * Patients with clinically significant cardiovascular disease. This includes: * Uncontrolled hypertension, defined as systolic greater than 140 mm Hg or diastolic greater than 90 mm Hg despite antihypertensive medications. * Myocardial infarction or unstable angina within 6 months prior to registration. * New York Heart Association (NYHA) Class II or greater congestive heart failure. (see Appendix III ) * History of serious ventricular arrhythmia (i.e., ventricular tachycardia or ventricular fibrillation) or serious cardiac arrhythmia requiring medication. This does not include asymptomatic atrial fibrillation with controlled ventricular rate. * Any history of congenital long QT syndrome * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \>500 ms within 28 days before randomization. Note: if initial QTcF is found to be \> 500 ms, two additional EKGs separated by at least 3 minutes should be performed. If the average of these three consecutive results for QTcF is ≤500 ms, the subject meets eligibility in this regard. * Patients with serious non-healing wound, ulcer, or bone fracture within 28 days before registration * Patients with history of organ transplant. * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving (in contact with, invading or encasing) major vessels. * Gastrointestinal disorders, particularly those with potential risk of perforation or fistula formation including: * Any of the following within 28 days of registration * Intra-abdominal tumor/metastases invading GI mucosa * Active peptic ulcer disease * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Malabsorption syndrome. * Any of the following within 6 months of registration * Abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction * Note: Patients requiring drainage gastrostomy (e.g., PEG tube) and/or parenteral hydration and/or nutrition are not eligible. * Intraabdominal abscess. * Note: Complete resolution of an intraabdominal abscess must be confirmed prior to registration even if the abscess occurred more than 6 months prior to registration. * Patients with history or evidence upon physical examination of CNS disease, including primary brain tumor, seizures which are not controlled with non-enzyme inducing anticonvulsants, any brain metastases and/or epidural disease, or history of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months prior to the first date of study treatment. * Major surgery within 3 months of the first dose of study drugs if there were no wound healing complications or within 6 months of the first dose of study drugs if there were wound complications. * The following are additional exclusion criteria for patients enrolling in Expansion Cohort C: * Uncontrolled blood pressure (\>140/90). Patients should have a blood pressure of ≤140/90 taken by a medical professional within one week of starting on study * Proteinuria \>2+ on urinalysis * Serosal involvement of the bowel that would render the patient at increased risk of gastrointestinal perforation