VEGF Trap in Treating Patients With Recurrent, Locally Advanced, or Metastatic Cancer of the Urothelium

Inclusion Criteria: * Histologically confirmed transitional cell carcinoma (TCC) of the urothelium * Must have predominance of transitional histology, but foci of squamous and/or adenocarcinoma histology allowed * Poorly differentiated transitional cell carcinoma allowed * TCC of any of the following sites allowed: * Bladder * Renal pelvis * Ureter * Urethra * Measurable disease, defined as \>= 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques OR \>= 10 mm by spiral CT scan * Locoregionally advanced or metastatic disease that is not amenable to curative surgery and/or radiotherapy * Must have received 1 prior systemic chemotherapy regimen containing a platinum compound (e.g., cisplatin, carboplatin, or oxaliplatin) in the neoadjuvant, adjuvant, or metastatic setting * No evidence of CNS disease, including primary brain tumor or brain metastases * ECOG performance status 0-2 * Absolute neutrophil count \>= 1,000/mm\^3 * Platelet count \>= 75,000/mm\^3 * Bilirubin =\< 1.5 times upper limit of normal (ULN) * AST or ALT =\< 2.5 times ULN * Creatinine =\< 2.5 times ULN OR creatinine clearance =\> 40 mL/min * Urine protein: creatinine ratio =\< 1 OR 24-hour urine protein \< 500 mg * INR =\< 1.5 (unless on full-dose warfarin) * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception during and for \>= 6 months after completion of study treatment * No known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies * No history of allergic reactions attributed to compounds of similar chemical or biological composition to other agents used in the study * No serious or nonhealing wound, ulcer, or bone fracture * No abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within the past 28 days * No significant traumatic injury within the past 28 days * No clinically significant cardiovascular disease, including any of the following: * Myocardial infarction, coronary artery bypass graft, or unstable angina pectoris within the past 6 months * New York Heart Association class III or IV congestive heart failure * Serious cardiac arrhythmia requiring medication * Clinically significant peripheral vascular disease within the past 6 months * Cerebrovascular accident within the past 6 months * Pulmonary embolism, deep vein thrombosis, or other thromboembolic event within the past 6 months * Uncontrolled hypertension, defined as blood pressure (BP) \> 150/100 mm Hg or systolic BP \> 180 mm Hg (if diastolic BP \< 90 mm Hg) within the past 3 months * No evidence of bleeding diathesis or coagulopathy * No uncontrolled intercurrent illness, including, but not limited to, any of the following: * Ongoing or active infection * Psychiatric illness or social situation that would preclude study compliance * Recovered from prior therapy * Prior biologic or targeted therapies allowed * No more than 1 prior systemic chemotherapy regimen for metastatic disease * No prior antiangiogenic therapy primarily targeting the vascular endothelial growth factor pathway * At least 4 weeks since prior radiotherapy or systemic therapy (6 weeks for mitomycin C or nitrosoureas) * More than 28 days since prior major surgery or open biopsy * More than 7 days since prior core biopsy * No concurrent major surgery * Concurrent full-dose anticoagulants (e.g., warfarin) allowed provided all of the following criteria are met: * In-range INR (usually between 2 and 3) on a stable dose of oral anticoagulant or low molecular weight heparin * No active bleeding or pathological condition that carries a high risk of bleeding (e.g., tumor involving major vessels or known varices) * No other concurrent investigational agents * No concurrent combination antiretroviral therapy for HIV-positive patients