Cabazitaxel With or Without Carboplatin in Treating Patients With Previously Treated Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria

• •Histologic evidence of prostate adenocarcinoma
• •In addition to patients with adenocarcinoma, patients with "anaplastic" features are also eligible as defined by at least one of the following: a) histologic evidence of small cell prostate cancer (patients with small cell carcinoma on histology are not required to demonstrate castration-resistant progression); b) any of the following metastatic presentations: (i) exclusive visceral metastases; (ii) radiographically predominant lytic bone metastases identified by plain X-ray or computed tomography (CT) scan; (iii) bulky (\>= 5 cm in longest dimension) lymphadenopathy (iv) bulky (\>= 5 cm) tumor mass in the prostate/pelvis (v) low PSA (=\< 10 ng/ml) at initial presentation (prior to androgen ablation or at symptomatic progression in the castrate-setting) plus high volume (\>= 20) bone metastases; (vi) elevated serum lactate dehydrogenase (LDH) (\>= 2 x ULN) or elevated serum carcinoembryonic antigen (CEA) (\>= 2 x upper limit of normal \[ULN\]) in the absence of other etiologies; (vii) short interval (=\< 180 days) to castrate-resistant progression following initiation of hormonal therapy
• •Castration-resistant prostate cancer; patients must have surgical or ongoing chemical castration (with luteinizing-hormone-releasing hormone \[LHRH\] agonists or LHRH antagonists), with a baseline testosterone level \< 50 ng/dL
• •Metastatic disease; patients must have evidence for metastatic prostate cancer by bone scan and/or CT/magnetic resonance imaging (MRI) (i.e., soft tissue, visceral, lymph node); if lymph node, visceral and/or soft-tissue metastases are the only evidence of metastasis, at least one lesion must be \>= 1.5 cm in diameter
• •Patients may have received prior treatment with androgen ablative therapies (such as bicalutamide, ketoconazole, diethylstilbestrol \[DES\], abiraterone, Xtandi, ARN-509) and/or "targeted" therapies (such as tyrosine kinase inhibitors); androgen ablative therapies must be discontinued \>= 3 days prior to initiation of study treatment with the exception of abiraterone and/or enzalutamide, which may be continued during study treatment per the practice preference of the treating physician; patients who are predicted to benefit from an antiandrogen withdrawal response should be tested for this possibility before being considered for eligibility to this study; targeted therapies must be discontinued \>= 2 weeks before initiation of study treatment
• •Both chemotherapy-naive and patients previously treated with chemotherapy are eligible; chemotherapy pretreated patients may have received a maximum of two prior systemic cytotoxic chemotherapies completed at least 3 weeks prior to initiation of study treatment
• •Patients must have documented evidence of progressive disease as defined by any of the following: a) PSA progression: minimum of 2 rising values (3 measurements) obtained a minimum of 7 days apart with the last result being at least \>= 2.0 ng/mL; b) new or increasing non-bone disease (by Response Evaluation Criteria In Solid Tumors \[RECIST\]); c) positive bone scan with 2 or more new lesions (Prostate Cancer Working Group \[PCWG2\])
• •For purposes of stratification, patients will be categorized as "responders" or "non-responders" based on their response to prior docetaxel-based therapy; a) responders will have demonstrated objective responses to first-line docetaxel as determined by any of the following: 1. decrease in PSA level \>= 50% from baseline, maintained for \>= 6 weeks; 2. partial or complete response in lymph nodes and soft tissue metastases by RECIST; responders must have received \>= 225 mg/m\^2 (\~ 3 cycles) of docetaxel; b) patients not meeting response criteria above will be considered as non-responders; we anticipate 2 general categories of non-responders based on the following disease phenotypes: 1. progressive disease on therapy without any objective evidence of response ("primary-resistant disease"); progressive disease on therapy with prior objective evidence of response, but response duration is =\< 6 weeks ("docetaxel refractory disease"); non-responders are eligible even if they have received \< 225 mg/m\^2 of docetaxel
• •If present, peripheral neuropathy must be =\< grade 2
• •Absolute neutrophil count (ANC) \>= 1,500/ml (unless due to bone marrow infiltration by tumor in which case ANC \>= 500/ml are allowed) (within 14 days before registration)
• •Platelets \>= 100,000/ml (unless due to bone marrow infiltration by tumor in which case \>= 50,000/ml are allowed) (within 14 days before registration)
• •Total bilirubin =\< upper limit of normal with the exception of isolated hyperbilirubinemia due to Gilbert's syndrome or if the patient has liver metastases and/or acute tumor-associated illness =\< 4 x ULN (within 14 days before registration)
• •Serum glutamic-pyruvic transaminase (SGPT), (alanine aminotransferase \[ALT\]) AND/OR serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 1.5 x the ULN or if patient has liver metastases and/or acute tumor-associated illness, =\< 4 x ULN (within 14 days before registration)
• •Patient has creatinine clearance \>= 30 ml/min using the Cockcroft-Gault equation (within 14 days before registration)
• •Men whose partner is a woman of childbearing potential must be willing to consent to using effective contraception while on treatment and for at least 3 months thereafter
• •Patient or his legally authorized representative must provide written informed consent
• •Eastern Cooperative Oncology Group (ECOG) performance status =\< 2