Anti-ICOS Monoclonal Antibody MEDI-570 in Treating Patients With Relapsed or Refractory Peripheral T-cell Lymphoma Follicular Variant or Angioimmunoblastic T-cell Lymphoma

Inclusion Criteria: * Pathologic diagnosis of one of the following: * For dose escalation: * Confirmed diagnosis of peripheral T-cell lymphoma (PTCL) or angioimmunoblastic T-cell lymphoma (AITL) that is refractory to at least one line of therapy; anaplastic large cell lymphoma (ALCL) and natural killer T-cell lymphoma nasal type (NKTCL) are excluded * Advanced stage cutaneous T-cell lymphoma (CTCL), specifically CTCL NOS, small/medium T-cell lymphoma (SMTCL) and mycosis fungoides (MF) stage IB, IIA, IIB, III and IV that have relapsed after at least one specific prior therapy (e.g. interferon, photopheresis, denileukin difitox, bexarotene, etc); anaplastic cutaneous large cell lymphoma (ACLCL) and lymphomatoid papulopsis are excluded * Follicular lymphoma grade 1, 2 or 3A that meets the following criteria: * Relapsed or refractory to at least 2 lines of therapy AND * Relapsed or refractory post autologous cell transplantation (HCT) * For dose expansion/dose confirmation phase: * Patients with confirmed diagnosis of peripheral T-cell lymphoma (PTCL) follicular type or angioimmunoblastic T-cell lymphoma (AITL) that is refractory to at least one line of therapy * At least 14 days from the last therapy dose or 5 half-lives (whichever is shorter), and resolution of toxicity related to the last therapy, excluding grade 2 or less peripheral neuropathy and alopecia; for radiation therapy, a minimum of 2 weeks and resolution of all acute toxicity will be required * Patients must have at least one measurable lesion that can be accurately measured with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI) scan, or physical exam (by calipers only); (PTCL, AITL and follicular lymphoma patients will be assessed on this study using the Lugano criteria for the evaluation of lymphomas; CTCL and MF patients will be assessed using International Society for Cutaneous Lymphomas \[ISCL\] and European Organization for Research and Treatment of Cancer \[EORTC\] criteria) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 6 months * Leukocytes \>= 3,000/mcL * Hemoglobin \>= 80 d/L (or \>= 8 g/dL) * Patients must not have received a transfusion, with packed red blood cells, within 2 weeks prior to sample being collected * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets (PLT) \>= 50,000/mcL * Absolute CD4 count \> 100 cells/uL * Total bilirubin \< 1.5 upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine \< 1.5 mg/dl (= 132 umol/L) or * Creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * In patients with bone marrow involvement the minimum requirement is as follows: * Leukocytes \>= 2000/mcL * ANC \>= 1000/mcL * PLT \>= 50 000/mcL * Availability of tissue for correlative studies; patients must have at least 6-8 unstained slides of archived formalin-fixed, paraffin-embedded tumor tissue available; if not enough archived tissue is available, a fresh tumor biopsy prior to study initiation is mandatory; for patients who have undergone a fresh baseline biopsy at baseline, the archived tissue is not mandatory * The effects of MEDI-570 on the developing human fetus are unknown; for this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, during the study participation, and for 3 months after the last dose of the drug; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must have either had a prior vasectomy or agree to use effective contraception prior to the study, during the study, and for 3 months after the last dose of the drug; males should avoid fathering children during and for at least three months after therapy is completed * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who are receiving any other investigational agents * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MEDI-570 or history of anaphylaxis to any biological component * Any history or evidence of opportunistic infection within 6 months of screening including tuberculosis, severe cytomegalovirus (CMV) or herpetic infections (such as disseminated herpes, herpes encephalitis, ophthalmic herpes) * Evidence of active infection by hepatitis B and/or C; active viral infection by hepatitis B and hepatitis C could be associated with cytopenias (due to hypersplenism or due to the active virus itself), which could add further risk when a potential immunosuppressive medication is used; for patients with hepatitis B treated with anti-virals to undetectable viral load, and for patients with hepatitis C with undetectable ribonucleic acid (RNA) levels and no evidence of liver damage, enrollment may be considered and should discuss first with study's principal investigator * History of human immunodeficiency virus (HIV) infection; the human immunodeficiency virus (HIV) depletes CD4 T-cells and could also have a role in T-cell anergy; since MEDI-570 preferentially affects CD4 T-cell numbers and function, and the resultant immunosuppression by this agent can be prolonged, exposing HIV patients to MEDI-570 will place them in an unnecessary risk of developing infections due to an underlying acquired cellular immunity defect * History of primary immunodeficiency * Receipt of live or live attenuated vaccine within 12 weeks prior to enrollment * All potential patients must undergo a tuberculosis (TB) test prior to study entry to rule out active or latent tuberculosis (either purified protein derivative \[PPD\] or QuantiFERON-TB Gold, whichever is preferred and available at the institution); patients with a history of TB (even if treated), or evidence of active or latent TB, are excluded; the diagnosis of active TB is defined per current guidelines; patients with a positive TB test (e.g. PPD or QuantiFERON-TB Gold) will be excluded; patients with history of Bacille-Calmette-Guerin (BCG) vaccination will be tested with QuantiFERON-TB Gold test in order to rule out exposure to TB * Patients who have undergone allogeneic stem cell transplantation * Patients who have undergone autologous stem cell transplantation within 3 months from study entry * Major surgery within 30 days prior or during the study period * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study due to the potential toxicity in pre-clinical reproductive studies; in addition, there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with MEDI-570; breastfeeding should be discontinued if the mother is treated with MEDI-570 * Patients with active, known, or suspected autoimmune disease, except in these conditions: * Participants with well-controlled asthma and/or mild allergic rhinitis (seasonal allergies) are eligible * Participants with the following disease conditions are also eligible: * Vitiligo, * Type 1 diabetes mellitus * Residual hypothyroidism due to autoimmune condition only requiring hormone replacement * Euthyroid participants with a history of Grave's disease (participants suspected autoimmune thyroid disorders must be negative for thyroglobulin and thyroid peroxidase antibodies and thyroid stimulating immunoglobulin prior to first dose of study drug) * For patients with ITP (idiopathic thrombocytopenic purpura) or AIHA (autoimmune hemolytic anemia), a case by case discussion with study principal investigator (PI) may be considered * Patients not receiving systemic therapy (i.e., systemic steroids or biologic therapy with disease modifying anti-rheumatic drugs \[DMARDs\]) within 2 years can be also eligible * Patients with a weight of \< 39 kg