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

Exclusion Criteria:

* Radiation therapy (including palliative radiotherapy to a metastatic lesion) within 14 days or major surgery (e.g., open abdominal, pelvic, thoracic, orthopedic or neurosurgery) within 28 days of the date of the first dose
* Samarium-153 within 28 days of registration, or strontium-89 within 12 weeks (84 days) of registration; patients who have received 2 or more doses of bone-seeking radioisotopes are not eligible
* Current treatment on another therapeutic clinical trial
* Prior treatment with cabazitaxel and/or carboplatin
* Impending complication from bone metastases (fracture and/or cord compression); properly treated or stabilized fractures and/or cord compression is allowed
* Presence of ongoing urinary obstruction (e.g., urinary retention, hydronephrosis) requiring medical intervention; properly treated urinary obstruction is allowed
* Patient has an uncontrolled intercurrent illness (e.g., uncontrolled diabetes, uncontrolled hypertension)
* Patient has another serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the patient's ability to provide informed consent or with the completion of treatment according to this protocol
* Patients with a history of severe hypersensitivity reaction to JEVTANA® (cabazitaxel) or other drugs formulated with polysorbate 80
* Patients with an active second malignancy that could, in the investigator's opinion, potentially interfere with the patient's ability to participate and/or complete this trial