Pembrolizumab, Endocrine Therapy, and Palbociclib in Treating Postmenopausal Patients With Newly Diagnosed Metastatic Stage IV Estrogen Receptor Positive Breast Cancer

Inclusion Criteria: * Willing and able to provide written informed consent/assent for the trial * Men or women \>= 18 years of age on day of signing informed consent * Willing and able to comply with all aspects of the treatment protocol * Postmenopausal patients defined by at least one of the following criteria: * Prior bilateral oophorectomy OR amenorrheic for \>= 12 months (if =\< 55 years of age and prior to chemotherapy, or on medical ovarian ablative therapy OR * Previous hysterectomy with one or both ovaries left in place (previous hysterectomy in which documentation of bilateral oophorectomy is unavailable AND follicle stimulating hormone \[FSH\] values consistent with the institutional normal values for the post-menopausal state; FSH levels must be obtained within 28 days prior to registration) * Presence of measurable disease meeting the following criteria: at least 1 lesion of \> 10 mm in long axis diameter for non-lymph nodes or \> 15 mm in short axis diameter for lymph nodes that is serially measurable according to RECIST version 1.1 using computerized tomography, magnetic resonance imaging, or panoramic and close-up color photography * Stage IV metastatic ER+HER2- breast cancer histologically proven per current American Society of Clinical Oncology (ASCO)/College of American Pathologists (CAP) guidelines; allow up to 30 days prior use of CDK4/6 inhibitors and up to 60 days of letrozole or other aromatase inhibitors for treatment of metastatic ER+ breast cancer * Life expectancy of \>= 3 months * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0-2 * Willing to provide tissue from a newly obtained core or excisional biopsy of a tumor lesion; newly-obtained is defined as a specimen obtained up-to 6 weeks (42 days) prior to initiation of treatment on day 1 and day -28 for cohort 3; subjects for whom newly-obtained samples cannot be provided (e.g. inaccessible or subject safety concern) may submit an archived specimen only upon agreement from the study principal investigator (PI) * For Cohort 3, willing to undergo tumor biopsies at baseline (within 6 weeks of study onset), cycle 2 day 1 (C2D1) (+/-1 week) and at time-of-progression or end-of-treatment when feasible * Absolute neutrophil count (ANC) \>= 1,000 /mcL (performed within 10 days of treatment initiation) * Platelets \>= 100,000 /mcL (performed within 10 days of treatment initiation) * Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment) (performed within 10 days of treatment initiation) * Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 60 mL/min for subject with creatinine levels \> 1.5 x institutional ULN (performed within 10 days of treatment initiation) * Creatinine clearance should be calculated per institutional standard * Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN (performed within 10 days of treatment initiation) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for subjects with liver metastases (performed within 10 days of treatment initiation) * Albumin \>= 2.5 mg/dL (performed within 10 days of treatment initiation) * International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Activated partial thromboplastin Time (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Female subjects of childbearing potential should 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 * Female subjects of childbearing potential should be willing to use two 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; subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year * Cohort 1 (accrual to 6 patients) is for patients who had ongoing stable disease (SD) on letrozole + palbociclib, enrolled on prior version of eligibility criteria to receive pembrolizumab after obtaining stable disease on letrozole + palbociclib; these patients must have been on treatment with letrozole and palbociclib for 6 months with SD per RECIST 1.1; received up to 3 lies of previous therapy including endocrine and/or chemotherapy in advanced setting prior to initiation of letrozole and palbociclib; no grade 3 toxicities except alopecia Exclusion Criteria: * Patients currently participating and receiving study therapy or who have participated in a study of an investigational agent and received study therapy or used and investigational device within 4 weeks of the first dose of treatment * Previously received pembrolizumab or other anti-programmed cell death-1 (PD-1) or anti-PD-L1 immunotherapy * Does not have measurable disease per RECIST 1.1 * For cohort 2, received \> 30 days of prior treatment with CDK4/6 inhibitors or \> 60 days of letrozole before screening * 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 * Known history of active TB (Bacillus tuberculosis) * Hypersensitivity to pembrolizumab or any of its excipients * Prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study day 1 or has not recovered (i.e., =\< grade 1 or at baseline) for adverse events (AEs) due to agents administered \> 4 weeks earlier * Prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study day 1 or has not recovered (i.e. =\< grade 1 or at baseline) from AEs due to a previously administered agent * Note: Patients with =\< grade 2 neuropathy are an exception to this criterion and may qualify for the study * Note: If patient received major surgery, she must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy * Known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer * Known active central nervous system (CNS) metastases and/or carcinomatous meningitis; patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least 4 weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment; this exception does not include carcinomatous meningitis, which is excluded regardless of clinical stability * Active autoimmune disease that has required systemic treatment in the past 2 years (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, etc.) is not considered a form of systemic treatment * History of (non-infectious) pneumonitis that required steroids or current pneumonitis * History of interstitial lung disease * Clinically active diverticulitis, intra-abdominal abscess, gastrointestinal (GI) obstruction, or abdominal carcinomatosis (known risks factors for bowel perforation) * Active infection requiring systemic therapy * History of significant cardiovascular disease, defined as: congestive heart failure greater than New York Heart Association (NYHA) class II according to the NYHA functional classification; unstable angina or myocardial infarction within 6 months of enrollment; or serious cardiac arrhythmia * Clinically significant electrocardiogram (ECG) abnormality, including a marked baseline prolonged QT/corrected QT (QTc) (\[QT interval/corrected QT interval\], e.g., a repeated demonstration of a QTc interval \> 480 ms), a family or personal history of long or short QT syndrome, Brugada syndrome or known history of QTc prolongation, or torsade de pointes (TdP) * Concurrent use of drugs that are known to be moderate or strong cytochrome P450, family 3, subfamily A (CYP3A) inhibitors or inducers or drugs that are known to prolong the QT interval * History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the patient's participation for the full duration of the trial, or is not in the best interests of the patient to participate, in the opinion of the treating investigator * Pregnant, breastfeeding, or expecting to conceive children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment * Known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial * Known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies) * Known active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected) * Received a live vaccine within 30 days of planned start of study therapy; * Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed