Romidepsin in Treating Patients With Locally Advanced or Metastatic Neuroendocrine Tumors

Inclusion Criteria: * Histologically confirmed carcinoid tumor or islet cell neuroendocrine tumor * Well- or moderately-differentiated tumor * Metastatic and/or locally advanced disease * Measurable disease * Unidimensionally measurable lesion at least 20 mm by conventional techniques OR at least 10 mm by spiral CT scan * Lesions in a previously irradiated area are not considered measurable * No truly non-measurable lesions, including the following: * Bone lesions * Leptomeningeal disease * Ascites * Pleural or pericardial effusion * Lymphangitis cutis/pulmonis * Abdominal masses not confirmed and followed by imaging * Cystic lesions * Ineligible for standard treatment * Performance status - ECOG 0-1 * At least 6 months * WBC \>= 3,000/mm\^3 * Absolute neutrophil count \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Bilirubin =\< 1.5 mg/dL * AST and ALT =\< 2.5 times upper limit of normal * Creatinine =\< 1.5 mg/dL * No New York Heart Association class III or IV congestive heart failure * No myocardial infarction within the past year * No uncontrolled dysrhythmias * No poorly controlled angina * No serious ventricular arrhythmia, defined as ventricular tachycardia or ventricular fibrillation \>= 3 beats in a row * No left ventricular hypertrophy by EKG * No other significant cardiac disease * QTc \< 500 msec * LVEF \> 40% by resting MUGA * No prior allergic reaction attributed to compounds of similar chemical or biological composition to study drug * No ongoing or active infection * No psychiatric illness or social situation that would preclude study compliance * No other concurrent uncontrolled illness * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * More than 4 weeks since prior immunotherapy (e.g., interferon alfa) * More than 4 weeks since prior chemotherapy * More than 12 weeks since prior hepatic artery chemoembolization unless liver lesions are not the only indicator lesions * No prior FR901228 (depsipeptide) * No more than 1 prior systemic chemotherapy regimen for carcinoid or islet cell tumor (other than hepatic artery chemoembolization) * More than 4 weeks since prior oral or IV steroids (first 16 patients only) * Concurrent long-acting octreotide allowed at standard doses if dose has been stable for the past 12 weeks * Concurrent subcutaneous octreotide for breakthrough use for symptomatic relief allowed * No concurrent systemic steroids (first 16 patients only) * More than 4 weeks since prior radiotherapy * More than 4 weeks since prior investigational tumor-specific therapy * No other prior histone deacetylase inhibitors (e.g., valproic acid) * No concurrent hydrochlorothiazide * No concurrent combination antiretroviral therapy for HIV-positive patients * No other concurrent investigational or commercial agents or therapies for the malignancy