Testing the Addition of an Antiangiogenic Drug (Bevacizumab) to Chemotherapy (Carboplatin and Paclitaxel) Combined With Immunotherapy (Pembrolizumab) for pMMR, TP53 Mutated Endometrial Cancer

Inclusion Criteria: * Documentation of disease: * Stage III and stage IVA endometrial cancers (with measurable disease), * Stage IVB endometrial cancer (with or without measurable disease), or * Recurrent endometrial cancer (with or without measurable disease) * In patients with measurable disease, lesions will be defined and monitored by RECIST 1.1. Measurable disease (RECIST 1.1) is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be ≥ 10 mm when measured by CT or MRI. Lymph nodes must be ≥ 15 mm in short axis when measured by CT or MRI * Histologic confirmation of the original primary tumor is required (submission of pathology report\[s\] is required). Patients with the following histologic types are eligible: endometrioid, serous, dedifferentiated/undifferentiated, clear cell, mixed epithelial, carcinosarcoma, adenocarcinoma not otherwise specified (N.O.S.) * Patients must have: * Tumoral mismatch repair proficient (pMMR) disease as assessed by immunohistochemistry (IHC) AND * P53 IHC with aberrant staining pattern (aberrant p53 expression is consistent with mutant TP53). TP53 mutation by next-generation sequencing will also be accepted * A pathology report demonstrating results of institutional MMR IHC and p53 IHC and/or TP53 by next-generation sequencing * Patients may have received: * NO prior chemotherapy for treatment of endometrial cancer OR * Prior adjuvant chemotherapy (e.g., paclitaxel/carboplatin alone or as a component of concurrent chemotherapy and radiation therapy \[with or without cisplatin\]) provided adjuvant chemotherapy was completed ≥ 12 months prior to registration * Patients may have received prior radiation therapy for treatment of endometrial cancer. Prior radiation therapy may have included pelvic radiation therapy, extended field pelvic/para-aortic radiation therapy, intravaginal brachytherapy, and/or palliative radiation therapy. All radiation therapy must be completed at least 4 weeks prior to registration. For patients with recent radiation, they must have RECIST-evaluable disease outside of the radiation field and have recovered their marrow function * Patients may have received prior hormonal (endocrine) therapy. All hormonal (endocrine) therapy must have been completed at least 1 week prior to registration * NO prior pembrolizumab (or other anti-PD1, anti-PDL1 or anti-CTLA4 therapy) or bevacizumab (or other antiangiogenic therapy) * Interval or cytoreductive surgery, after start of treatment on this trial, and prior to documentation of disease progression, is NOT permitted * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of disease progression. Patients with brain metastases must have follow up imaging demonstrating no evidence of disease progression and that the disease is stable off of steroids * Age ≥ 18 * Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 2 * Not pregnant and not nursing * Absolute neutrophil count (ANC) ≥ 1,500 cells/mm\^3 * Platelets ≥ 100,000 cells/mm\^3 * Hemoglobin ≥ 8 g/dl * Creatinine clearance (CrCl) of ≥ 30 mL/min by the Cockcroft-Gault formula * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (patients with known Gilbert's disease who have bilirubin level ≤ 3 x institutional ULN may be enrolled) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 x institutional ULN * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better * No active infection requiring parenteral antibiotics * No current evidence of intra-abdominal abscess, abdominal/pelvic fistula (not diverted), gastrointestinal perforation, gastrointestinal (GI) obstruction, and/or need for drainage nasogastric or gastrostomy tube * No clinically significant bleeding within 28 days prior to registration * No uncontrolled hypertension, defined as systolic ≥ 160 mm Hg or diastolic ≥ 100 mm Hg * No major surgery within 28 days of initiation of bevacizumab * No active autoimmune disease or history of autoimmune disease that might recur, which may affect vital organ function or require immune suppressive treatment including corticosteroids. This includes, but is not limited to, patients with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as systemic lupus erythematosus (SLE), connective tissue diseases, scleroderma, inflammatory bowel disease (IBD), Crohn's, ulcerative colitis, hepatitis; and patients with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome because of the risk of recurrence or exacerbation of disease * Patients with vitiligo, endocrine deficiencies including type I diabetes mellitus, thyroiditis managed with replacement hormones including physiologic corticosteroids are eligible * Topical or inhaled steroids are allowed * Patients with rheumatoid arthritis and other arthropathies, Sjogren's syndrome and psoriasis controlled with topical medication and patients with positive serology, such as antinuclear antibodies (ANA), and anti-thyroid antibodies should be evaluated with the presence of target organ involvement and potential need for systemic treatment but should otherwise be eligible * No history of (non-infectious) pneumonitis that required steroids, or current pneumonitis * No history of stem cell or solid organ transplant * No history of allergic reaction to the study agent(s) or compounds of similar chemical or biologic composition to the study agent(s) (or any of its excipients)