APL-2 and Pembrolizumab Versus APL-2, Pembrolizumab and Bevacizumab Versus Bevacizumab Alone for the Treatment of Recurrent Ovarian, Fallopian Tube, or Primary Peritoneal Cancer and Malignant Effusion

Inclusion Criteria: * Age \>= 18 years of age on day of signing informed consent * Recurrent epithelial ovarian/fallopian tube or primary peritoneal cancer (serous, clear cell, endometrioid, mixed or poorly differentiated or carcinosarcoma) based on imaging or synchronous primary ovarian and uterine cancer patients with any of the histology subtypes mentioned above regardless of platinum sensitivity, prior stage or number of prior treatment lines * Symptomatic ascites or pleural effusion or both requiring \>= 1 drainage within 4-weeks of study entry or has a peritoneal/pleural drainage catheter in place to control symptoms * Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Patient has not received pembrolizumab or other immune checkpoint inhibitor treatment for 9 weeks prior to enrollment * Life expectancy of \>= 3 months * Absolute neutrophil count (ANC): \>= 1,500/µL * Platelets: \>= 75,000/µL * Hemoglobin: \>= 9 g/dL or 5.6 mmol/L (within 7 days of assessment) * Creatinine: =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance \>= 60 mL/min (Cockcroft-Gault Equation) for participant with creatinine levels \> 1.5 X institutional ULN. GFR can also be used in place of creatinine or creatinine clearance (CrCl) * Total bilirubin: =\< 1.5 X ULN OR direct bilirubin =\< ULN for participants with total bilirubin levels \> 1.5 ULN * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]): =\< 2.5 X ULN OR =\< 5 X ULN for participants with liver metastases * Albumin: \> 2.5 gm/dL * International Normalized Ratio (INR) or Prothrombin Time (PT): =\< 1.5 unless participant is receiving anticoagulant therapy as long as PT or activated partial thromboplastin time (aPTT) is within therapeutic range of intended use of anticoagulants * Activated Partial Thromboplastin Time (aPTT): =\< 1.5 X ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants * A woman of childbearing potential must have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Participants of childbearing potential must be willing to use 2 methods of birth control or be surgically sterile or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication (participants of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year). Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately * Willing and able to self-administer APL-2 (administration by caregiver will be allowed) * No known absolute contraindication to bevacizumab and/or pembrolizumab treatment per enrolling provider * Willing to receive vaccination against Neisseria meningitidis, Streptococcus pneumoniae, and Hemophilus influenzae if randomized into an APL-2 receiving arm, if not already vaccinated * Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure Exclusion Criteria: * Is currently receiving any additional cancer therapy or participating or used an investigational drug or device within 3 weeks of the first dose of treatment * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment or, is taking any other medication that might affect immune function * Has active autoimmune disease that has required systemic treatment in the past 3 months (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) is not considered a form of systemic treatment * Has an active infection requiring systemic therapy * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the participant's participation for the full duration of the trial, or is not in the best interest of the patient to participate, in the opinion of the treating investigator * Participant has clinically significant cardiovascular disease including: * Uncontrolled hypertension, defined as systolic \>150 mmHg or diastolic \>90 mmHg * Myocardial infarction or unstable angina within 6 months prior to enrollment * New York Heart Association (NYHA) Grade II or greater congestive heart failure * Participant has a Grade II (NYHA) or greater peripheral vascular disease * Participant has a clinically significant peripheral artery disease (e.g. those with claudication), within 6 months prior to study enrollment * Pregnancy or lactation * Unwilling or unable to follow protocol requirements * Any condition which in the investigator's opinion deems the participant an unsuitable candidate to receive study drug * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Has a known history of human immunodeficiency virus (HIV) infection * Concurrent active hepatitis B (defined as hepatitis B surface antigen \[HBsAg\] positive and/or detectable HBV deoxyribonucleic acid \[DNA\]) and hepatitis C virus (HCV) (defined as anti-HCV Ab positive and detectable HCV ribonucleic acid \[RNA\]) infection. Note: Hepatitis B and C screening tests are not required unless known history of HB