Activated T-cell Therapy, Low-Dose Aldesleukin, and Sargramostim in Treating Patients With Ovarian, Fallopian Tube, or Primary Peritoneal Cancer That is Stage III-IV, Refractory, or Recurrent

Inclusion Criteria:

* Histologically documented, epithelial ovarian, fallopian tube, or primary peritoneal, high grade serous or clear cell carcinoma are eligible; all patients must have a confirmed pathology; stage 3 and 4 initial disease with response to primary surgery and neo/adjuvant chemotherapy, platinum refractory disease, and patients with recurrent disease are candidates
* Patients meeting the above pathologic criteria will be eligible for therapy irrespective of their HER2/neu over expression status; immunohistochemical staining will be not be required for protocol entry but fluorescence in situ hybridization (FISH) and immunohistochemistry (IHC) studies for HER2/neu are preferred
* Chemotherapy: no limit to prior therapies; however, patients with multiple chemotherapy regimens will be screened for lymphocyte proliferation at investigator's discretion
* Herceptin: women who have been previously treated with Herceptin or other monoclonal antibody therapies are eligible for the trial
* Radiation therapy: patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis 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 may have no evidence of measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) criteria or have measurable disease; CA-125 and other available markers will be obtained
* Karnofsky performance score of \>= 70 is required or Eastern Cooperative Oncology Group (ECOG) score, performance status (PS) = 0-2
* The patient must have a life expectancy of 3 months or more based on the judgment of the investigators; women who have rapidly progressive symptomatic disease affecting major organ systems such as the liver and lungs will be excluded
* Negative serum test for pregnancy in premenopausal women
* No previous or concurrent malignancy, other than curatively treated in situ squamous cell carcinoma of the cervix or basal cell carcinoma of the skin or non-active breast cancer
* Each patient must be aware of the nature of her disease process and must willingly consent to treatment after being informed of alternatives, potential benefits, side effects, and risks; eligibility testing that is considered standard of care may be done prior to informed consent but no immunotherapy related procedures or testing may occur without informed consent
* No serious medical or psychiatric illness which prevents informed consent or intensive treatment
* Patients will be ineligible for treatment on this protocol if:

  * There is a history of a recent myocardial infarction (within one year)
  * There is a history of a past myocardial infarction (more than one year ago) along with current coronary symptoms requiring medications and/or evidence of depressed left ventricular function (left ventricular ejection fraction \[LVEF\] \< 45% by echocardiogram \[ECHO\])
  * There is a current history of angina/coronary symptoms requiring medications and/or evidence of depressed left ventricular function (LVEF \< 45% by ECHO)
  * There is clinical evidence of congestive heart failure requiring medical management (irrespective of ECHO results)
* Patients who have persistently elevated systolic blood pressures (BPs) \>= 145 or diastolic BPs \>= 90 need to have their systolic or diastolic BP controlled with anti-hypertensive agents for at least 3 days prior to the initiation of cell therapy; patients already on anti-hypertensive agents will have their medicine adjusted based on the clinical judgment of the patient care team
* Patients with treated brain metastases (received definitive radiation and/or underwent surgical resection) are eligible for therapy on this protocol; patients with clinical evidence of active brain metastases are ineligible for therapy on this protocol
* Granulocytes \>= 1,000/mm\^3
* Platelet count \>= 50,000/ul
* Hemoglobin \>= 8 gm/dl
* Blood urea nitrogen (BUN) =\< 1.5 times normal
* Serum creatinine =\< 1.8 mg/dl
* Creatinine clearance \>= 60 ml/mm
* Bilirubin =\< 2.0 times normal
* Serum glutamic oxaloacetic transaminase (SGOT) =\< 2.0 times normal
* Human immunodeficiency virus (HIV) = negative
* LVEF \>= 45% at rest (by ECHO)
* Pulmonary function tests (PFT)-forced expiratory volume in one second (FEV1), diffusing capacity of the lung for carbon monoxide (DLCO2), and forced vital capacity (FVC) \>= 60% predicted value if clinically indicated
* Minor changes from the required initial laboratory data guidelines will be allowed at the discretion of the attending team under special circumstances; the reasons for exceptions must be documented prior to enrollment
* Appropriate slides of the primary lesion will be available for future review; if available, HER2/neu positivity will be recorded
* Peritoneal dialysis catheter implantation is identical to that from the Gynecologic Oncology Group (GOG) 252 Protocol and revised from the GOG Surgical Procedures Manual