Inclusion Criteria:
1. Age ≥ 18 years.
2. Histologically confirmed classic Hodgkin lymphoma (including nodular sclerosis, lymphocyte-rich, mixed cellularity, and lymphocyte-depleted Hodgkin lymphoma).
3. Participants must have relapsed or refractory disease after no more than one line of systemic therapy. First line therapy must have included doxorubicin.
4. At least one site of FDG-avid disease on PET/CT that is ≥ 1.5 cm in diameter for nodal disease or ≥ 1.0 cm in diameter if extranodal disease.
5. Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 (Karnofsky ≥ 50%).
6. Demonstrates adequate organ function as defined below:
1. Absolute neutrophil count (ANC) ≥ 1.0 X 10\^9/ L (1000/microliter (mcL), growth factors permitted).
2. Platelets ≥ 75 X 10\^9 / L (75,000/mcL, platelet transfusion independent).
3. Total bilirubin ≤ 1.5 x institutional upper limit of normal, unless elevated due to Gilbert's syndrome.
4. Aspartate aminotransferase (AST) / (SGOT) ≤3 x institutional upper limit of normal.
5. Alanine aminotransferase (ALT) / (SGPT) ≤3 x institutional upper limit of normal.
6. Creatinine clearance (CrCl, calculated) ≥ 40 mL/min/1.73 m\^2, calculated using the Cockcroft-Gault equation.
7. For participants with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated.
8. Individuals with a history of hepatitis C virus (HCV) infection must have been treated with sustained virologic response. For individuals with HCV infection who are currently on treatment, participants are eligible if there is an undetectable HCV viral load.
9. Human immunodeficiency virus (HIV)-infected individuals on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial.
10. Individuals with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial.
11. Females of reproductive potential (defined below) must be willing to undergo a urine or serum pregnancy test (i.e., human chorionic gonadotropin test) within 72 hours before start of trial therapy. A female is considered to not be of reproductive potential (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice), if the participants meet either of the following two criteria: (1) has reached a postmenopausal state (≥ 12 continuous months of amenorrhea with no identified cause other than menopause); or (2) has undergone surgical sterilization (i.e., hysterectomy and/or bilateral oophorectomy for removal of uterus and/or ovaries). The result of the urine or serum pregnancy test must be negative in order to administer initiate trial therapy. If a urine pregnancy test cannot be confirmed as negative (e.g., an ambiguous result), a serum pregnancy test is required. In such cases, the individual must be excluded from participation if the serum pregnancy result is positive. Pregnant individuals are excluded from this study because there is a potential risk for adverse effects in the unborn child secondary to treatment of the study participant with trial therapy.
12. Ability to understand and willingness to sign a written informed consent document.
Exclusion Criteria:
1. Participants who previously received a programmed cell death protein 1 (PD-1) inhibitor-based regimen (e.g., nivolumab or pembrolizumab + AVD) and progressed within 6 months of last done of immune checkpoint inhibitor.
2. Receipt of systemic anti-cancer therapies or radiation within 2 weeks prior to the start of trial therapy or receipt of antibody therapy within 4 weeks prior to the start of trial therapy.
3. Has participated in a study of an investigational product and received study treatment or used an investigational device within four weeks of the first dose of treatment.
4. Prior autologous or allogeneic hematopoietic stem cell transplant.
5. Systemic autoimmune disease requiring continuous immunosuppressive treatment (≥prednisone 10 mg per day or equivalent), with the exception of autoimmune thyroid disease.
6. Left ventricular ejection fraction (LVEF) \<50% as assessed by transthoracic echocardiogram or multigated acquisition (MUGA) scan.
7. Participants whose lifetime cumulative dose of doxorubicin would exceed 450 mg/m\^2 after receiving 4 cycles of pembro + GVD are excluded (i.e., \>330 mg/m\^2 at trial entry).
8. Has known hypersensitivity to pembrolizumab, gemcitabine, vinorelbine, and/or liposomal doxorubicin; or any of their excipients.
9. Has any significant medical condition or comorbidity that could compromise participants safety in the opinion of the treating investigator (e.g., uncontrolled serious infection).
10. Individuals who are pregnant or breast-feeding/chest-feeding. Pregnant and breastfeeding individuals are excluded because there is a potential risk for adverse effects in the unborn/nursing child secondary to treatment of the study participant with trial therapies. Females of reproductive potential must have a negative pregnancy test before initiation of trial therapy, as outlined in inclusion criterion #12. Breast-feeding should be discontinued before initiation of trial therapy.
11. Individuals with ongoing Grade 2 events that are not clinically stable or ongoing ≥ Grade 3 events (CTCAE v5.0 grading).
12. Notification from Foresight Diagnostics that the CLARITY LDT baseline specimen has failed sample quality control (QC) and/or the baseline ctDNA is not quantifiable. Note: Foresight will only notify sites if the specimen fails quality control (QC) or is not quantifiable. Sites will not receive notification if the specimen passes QC and is quantifiable. For QC failures, a sample re-draw may be considered upon discussion with and approval by the lead University of California, San Francisco (UCSF) Principal Investigator.
13. Individuals with any condition or social circumstance that, in the opinion of the investigator, would impair the participant's ability to comply with study activities, interfere with participant safety, or study endpoints.
