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