Ipilimumab in Treating Patients With Metastatic or Recurrent Human Papilloma Virus-Related Cervical Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed metastatic or recurrent cervical cancer of squamous, adenocarcinoma or mixed histology type not suited to definitive localized therapy; HPV status will be confirmed for all patients following enrollment * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \> 10 mm with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Previous therapy: * Patients may have undergone surgery and/or received definitive radiation or chemo-radiation for localized disease in the past * Radiation treatment with curative intent (radical chemoradiotherapy or adjuvant chemoradiotherapy) must have been completed \>= 3 months prior to enrollment * Note: patients who completed palliative radiation therapy 2 weeks before start of ipilimumab are allowed as long as this does not affect measurable disease * Patients must have been exposed to platinum chemotherapy either as part of definitive chemo-radiation OR as first line systemic treatment for metastatic disease * Patients MAY have received up to two prior lines of systemic chemotherapy for metastatic or recurrent disease; patients with metastatic disease at first presentation MUST have received one platinum based line of chemotherapy * All chemotherapy must have been completed \>= 4 weeks prior to enrollment with radiologic evidence of radiological disease progression * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Life expectancy of greater than 3 months * Leukocytes \>= 3.0 x 10\^9/L * Absolute neutrophil count \>= 1.5 x 10\^9/L * Platelets \>= 100 x 10\^9/L * Total bilirubin within normal institutional limits (except in Gilbert's syndrome) * Thyroid stimulating hormone (TSH) =\< upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine \< 1.25 ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 as calculated by the Cockcroft and Gault formula * All radiology studies must be performed =\< 3 weeks prior to the start of therapy * Subjects with treated and asymptomatic brain metastases are eligible; patients that received palliative radiation (for brain metastases) are eligible if they have been asymptomatic for at least 2 weeks with use of maintenance steroid therapy, and last received radiation at least 4 weeks prior to start of therapy * Ability to understand and willing to sign a written informed consent document * Ongoing prior toxicities related to previous treatments must be recovered to =\< grade 1 at the time of registration (with the exception of alopecia or skin depigmentation) * Patients are willing to undergo tumor biopsy pre-treatment (within 14 days prior to registration) and post-treatment (within the first week of cycle 2 onset); patients who consent but have tumor that is not amenable to safe biopsy will be allowed to enter the trial/continue therapy as per protocol if this has been addressed and permission is granted from the lead consortium principal investigator (PI) prior to registration continuation of treatment Exclusion Criteria: * Patients who have had chemotherapy \< 4 weeks prior to enrollment (\< 6 weeks for nitrosoureas or mitomycin C) or who had radiation therapy with curative intent \< 3 months prior to enrollment (\< 2 weeks for palliative radiation therapy) or those who have not recovered (=\< grade 1) from adverse events related to previous treatments are excluded * Patients with a history of prior treatment with ipilimumab or other cytotoxic T-lymphocyte antigen 4 (CTLA4) agonists or antagonists, anti-programmed death 1 (PD 1) antibody, cluster of differentiation (CD)137 agonist or other immune activating therapy such as anti-CD 40 antibody are excluded * Patients who are receiving any other investigational agents * Autoimmune disease: patients with a history of inflammatory bowel disease, including ulcerative colitis and Crohn's disease, are excluded from this study, as are patients with a history of symptomatic disease (e.g., rheumatoid arthritis, systemic progressive sclerosis \[scleroderma\], systemic lupus erythematosus, autoimmune vasculitis \[e.g., Wegener's granulomatosis\]); central nervous system (CNS) or motor neuropathy considered of autoimmune origin (e.g. Guillain-Barre syndrome and Myasthenia Gravis, multiple sclerosis) * Patients requiring immunosuppressive agents, unless required for treating potential immune related adverse effects; steroids at their lowest effective dose in patients with radiated brain metastases is permitted * Patients with known immune impairment who may be unable to respond to anti-CTLA 4 antibody * Any other prior malignancy from which the patient has been disease free for less than 3 years, with the exception of adequately treated and cured basal or squamous cell skin cancer, superficial bladder cancer, carcinoma in situ of any site or any other cancer * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ipilimumab * Patients requiring systemic steroids are excluded; narcotics should be used with caution * 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; patients with chronic hepatitis B or hepatitis C infections should be excluded * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with ipilimumab * Patients with active or chronic infection with human immunodeficiency virus (HIV) who have raised viral loads or uncontrolled disease are ineligible; those patients however who exhibit minimal viral loads with good control whilst on stable anti-viral regimen may be considered if they meet all other eligibility criteria