Sorafenib With or Without Paclitaxel and Carboplatin in Treating Patients With Recurrent Ovarian Cancer, Primary Peritoneal Cancer, or Fallopian Tube Cancer

Inclusion Criteria: * Diagnosis of ovarian epithelial, primary peritoneal, or fallopian tube cancer * Recurrent disease * Must have received a prior platinum-based regimen * Platinum-sensitive (treatment-free interval \> 6 months) * No more than 2 prior chemotherapy regimens * Measurable disease * At least 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques OR \>= 10 mm by spiral CT scan * Not in a prior irradiation field * No known brain metastases * Performance status: * ECOG 0-2 OR * Karnofsky 80-100% * Life expectancy: * More than 12 weeks * Hematopoietic: * Absolute neutrophil count \>= 1,500/mm3 * Platelet count \>= 100,000/mm3 * Hemoglobin \>= 9 g/dL * No bleeding diathesis * Hepatic: * Bilirubin \< 1.5 times upper limit of normal (ULN) * AST or ALT =\< 2 times ULN * No history of allergic reaction attributed to compounds of similar chemical or biological composition to sorafenib or other agents used in the study * Patients who have had a reaction to a taxane or a platinum and have not yet been rechallenged may undergo a desensitization regimen on study * No hypersensitivity to paclitaxel or drugs using the vehicle Cremophor El: * Prior hypersensitivity reaction to paclitaxel allowed provided rechallenged successfully * Renal: * Creatinine \< 2 mg/dL * Cardiovascular: * Abnormal cardiac conduction (e.g., bundle branch block or heart block) allowed if stable for the past 6 months * No symptomatic congestive heart failure * No uncontrolled hypertension * No cardiac arrhythmia * No unstable angina pectoris; * No myocardial infarction within the past 6 months * Negative pregnancy test * Fertile patients must use effective contraception * Adequate intestinal function * No concurrent requirements for IV hydration or nutritional support * No active or ongoing infection * No psychiatric illness or social situation that would preclude study compliance * No other concurrent uncontrolled illness * No other invasive malignancy with the past 5 years except nonmelanoma skin cancer * More than 4 weeks since prior chemotherapy (6 weeks for nitrosoureas or mitomycin) and recovered * More than 3 weeks since prior hormonal therapy * More than 4 weeks since prior radiotherapy and recovered * No prior sorafenib * No prior anticancer therapy that contraindicates study therapy * No concurrent cytochrome P450 enzyme-inducing antiepileptic drugs (phenytoin, carbamazepine, or phenobarbital), rifampin, or Hypericum perforatum (St. John's wort) * No concurrent combination antiretroviral therapy for HIV-positive patients * No concurrent therapeutic anticoagulation therapy * Concurrent prophylactic low-dose warfarin allowed for maintenance of venous or arterial access devices * No other concurrent anticancer therapies * No other concurrent investigational agents * Not pregnant or nursing