Pegylated Liposomal Doxorubicin Hydrochloride, Bortezomib, Cyclophosphamide, and Dexamethasone in Treating Patients With Multiple Myeloma

Inclusion Criteria:

* Cohort 1: Relapsed, refractory patients with multiple myeloma who have failed at least one prior regimen not including dexamethasone alone
* Cohort 2: Newly diagnosed patients with previously untreated multiple myeloma; prior dexamethasone permitted; not to exceed 320 mg
* Diagnosis of multiple myeloma with quantifiable monoclonal protein or light chain identified by serum protein electrophoresis (SPEP), urine protein electrophoresis (UPEP), or serum free light chain assay
* Eastern Cooperative Oncology Group (ECOG) performance status 0-2
* Absolute neutrophil count \>= 1.5
* Platelet count \>= 75,000 unless slightly lower due to disease with the approval of the principal investigator (PI)
* Serum creatinine =\< 2.0 mg/dL
* Serum bilirubin =\< 1.2
* Aspartate transaminase (AST)/alanine transaminase (ALT) =\< 2.5 x upper limit of normal (ULN)
* Alkaline phosphatase =\< 2.5 x ULN
* Voluntary written informed consent before performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care
* Left ventricular ejection fraction greater than or equal to 50% by multi gated acquisition scan (MUGA)
* Female subjects must be post-menopausal, surgically sterilized, or willing to use an acceptable method of birth control (i.e., a hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study; female patients of childbearing potential must have a negative serum pregnancy test within 2 weeks prior to enrollment
* Male patients must use an effective contraceptive method during the study and for a minimum of 6 months after study treatment

Exclusion Criteria:

* Current concomitant chemotherapy, radiation therapy, or immunotherapy other than as specified in the protocol
* Use of other anticancer therapy within 15 days or before study entry; the patient must have recovered from all acute non-hematological toxicities from any previous therapy
* Have any other severe concurrent disease, or have a history of serious organ dysfunction or disease involving the heart, kidney, liver, or other organ system that may place the patient at undue risk to undergo treatment, despite appropriate antibiotics or other treatment; for cardiac dysfunction, myocardial infarction within 6 months prior to enrollment or New York Heart Association (NYHA) Class III or IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities
* Patients with a systemic fungal, bacterial, viral, or other infection not controlled (defined as exhibiting ongoing signs/symptoms related to the infection) on antiviral, antibiotic and antifungal treatment
* Patient has \>= Grade 2 peripheral neuropathy within 14 days before enrollment
* Patient has hypersensitivity to bortezomib, boron or mannitol
* Pregnant or lactating patients
* Cumulative dose of doxorubicin of 400 mg/m\^2 or greater, or if this level would be exceeded during the current study
* Any significant concurrent illness, condition, or psychiatric disorder that would compromise patient safety or compliance, interfere with consent, study participation, follow up, or interpretation of study results
* Have had a diagnosis of another malignancy, unless the patient has been disease-free for at least 3 years following the completion of curative intent therapy including the following:
* Patients with treated non-melanoma skin cancer, in situ carcinoma, or cervical intraepithelial neoplasia, regardless of the disease-free duration, are eligible for this study if definitive treatment for the condition has been completed;
* Patients with organ-confined prostate cancer with no evidence of recurrent or progressive disease based on prostate-specific antigen (PSA) values are also eligible for this study if hormonal therapy has been initiated or a radical prostatectomy has been performed;
* Prior autologous stem cell transplant (Cohort 2 only);
* Prior allogeneic stem cell transplant;
* Patient has received other investigational drugs within 14 days before enrollment