Bevacizumab and Combination Chemotherapy in Treating Patients With Peripheral T-Cell Lymphoma or Natural Killer Cell Neoplasms

INCLUSION CRITERIA: * Diagnosis of peripheral T-cell or natural killer cell neoplasm * Any stage disease allowed * HTLV-positive tumors allowed * At least one objective measurable disease parameter. Abnormal positron emission tomography scans are not considered evidence of measurable disease unless results are confirmed by CT scan or other appropriate imaging techniques * Age 18 and over * ECOG Performance status 0-2 * Absolute neutrophil count ≥ 1,000/mm\^3(500/mm\^3 if due to bone marrow involvement with lymphoma) * Platelet count ≥ 100,000/mm\^3(50,000/mm\^3 if due to bone marrow involvement with lymphoma) * Bilirubin ≤ 2.0 mg/dL (≤ 3 times upper limit of normal \[ULN\] if due to hepatic involvement with lymphoma) * AST ≤ 2 times ULN (5 times ULN if due to hepatic involvement with lymphoma) * PT, INR, and PTT ≤ 1.5 times normal * Creatinine ≤ 2.0 mg/dL * Urinary protein:creatinine ratio ≤ 1 * History of deep venous thrombosis allowed provided patient is on a stable dose of anticoagulants for at least 2 weeks prior to study entry * LVEF ≥ 50% * History of pulmonary embolism allowed provided patient is on a stable dose of anticoagulants for at least 2 weeks prior to study entry * One prior cycle of CHOP for PTCL allowed * More than 4 weeks since prior major invasive surgery or open biopsy * At least 7 days since prior minor surgery. Peripheral lymph node core biopsy, bone marrow biopsy, fine needle aspiration, skin biopsy, or central line placement are not considered minor surgical procedures * More than 7 days since prior and no concurrent anti-platelet drugs (e.g., ticlopidine, clopidogrel, or cilostazol) except aspirin or other nonsteroidal anti-inflammatory drugs * Concurrent anticoagulants allowed provided patient is on a stable dose * INR must be stable for at least 2 weeks prior to study entry * PT/INR and/or PTT must be closely monitored and levels kept within acceptable range for underlying thrombotic disease * Concurrent heparin flush for maintenance of central line patency allowed EXCLUSION CRITERIA: * Anaplastic lymphoma kinase (ALK)-positive T-cell large cell lymphoma. ALK-negative T-cell large cell lymphoma allowed * Cutaneous T-cell lymphoma * History of or current radiographic evidence of CNS metastasis, including previously treated, resected, or asymptomatic brain lesions or leptomeningeal involvement * Evidence of bleeding diathesis or coagulopathy * Cerebrovascular accident within the past 6 months * Myocardial infarction within the past 6 months * Unstable angina within the past 6 months * New York Heart Association class II-IV congestive heart failure * Uncontrolled hypertension (i.e., systolic blood pressure \[BP\] \> 150 mm Hg or diastolic BP \> 100 mm Hg) * Other clinically significant cardiovascular or peripheral vascular disease * Abdominal fistula within the past 6 months * Gastrointestinal perforation within the past 6 months * Intra-abdominal abscess within the past 6 months * Concurrent major surgery * Pregnant or nursing. Female patients must have negative pregnancy test. Fertile patients must use effective contraception * History of active seizures * Significant traumatic injury within the past 4 weeks * Non-healing ulcer (unless involved with lymphoma) * Bone fracture * Active infection requiring parenteral antibiotics * HIV positivity * Other active malignancy within the past 6 months except carcinoma in situ of the cervix or basal cell carcinoma of the skin