Abiraterone Acetate and Antiandrogen Therapy With or Without Cabazitaxel and Prednisone in Treating Patients With Metastatic, Castration-Resistant Prostate Cancer Previously Treated With Docetaxel

Inclusion Criteria:

* Histologically confirmed diagnosis of prostate cancer (adenocarcinoma of the prostate)
* Previous chemotherapy with at least 3 cycles of docetaxel for hormone-sensitive metastatic prostate cancer
* Metastatic disease as evidenced by the presence of soft tissue and/or bone metastases on imaging studies (CT/magnetic resonance imaging \[MRI\] of abdomen/pelvis, bone scintigraphy or NaF PET/CT)
* Ability to swallow abiraterone acetate tablets as a whole
* All patients must be receiving standard of care androgen deprivation treatment (surgical castration versus LHRH agonist or antagonist treatment); subjects receiving LHRH agonist or antagonist must continue treatment throughout the time on this study
* Patients must have castrate serum level of testosterone of \< 50 ng/dL (\< 1.73 nmol/L)
* Patients must have progressive disease while receiving androgen deprivation therapy defined by any one of the following as per the Prostate Cancer Clinical Trials Working Group 3 (PCWG3) criteria for PSA, measurable disease or non-measurable (bone) disease during treatment with ADT:

  * PSA: At least two consecutive rises in serum PSA, obtained at a minimum of 1-week intervals, with the final value \>= 1.0 ng/mL, confirmed \<=4 weeks prior to randomization
  * Measurable disease (by RECIST 1.1): \> 20% increase in the sum of the longest diameters of all measurable lesions or the development of new measurable lesions; the short axis of a target lymph node must be more than 15 mm to be assessed for change in size
  * Non-measurable (bone) disease: The appearance of two or more new areas of uptake on bone scan (or NaF PET/CT) consistent with metastatic disease compared to previous imaging during castration therapy; the increased uptake of pre-existing lesions on bone scan will not be taken to constitute progression, and ambiguous results must be confirmed by other imaging modalities (e.g. X-ray, CT or MRI). Clinical decisions about response or progression will be based on CT and bone scans.
* Patients may or may not have been treated previously with a nonsteroidal antiandrogen, such as flutamide, bicalutamide or nilutamide; for patients previously treated with an antiandrogen, they must be off treatment for at least 4 weeks (for flutamide) or 6 weeks (for bicalutamide or nilutamide) prior to registration and must have shown PSA progression after discontinuing the anti-androgen
* Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2
* Absolute neutrophil count (ANC) \>= 1500/mm\^3
* Hemoglobin (HgB) \>= 9.0 gr/dL
* Platelets \>= 100,000/mm\^3
* Creatinine \< 2.0 mg/dL
* Patients must be informed of the experimental nature of the study and its potential risks, and must sign an Institutional Review Board (IRB)-approved written informed consent form indicating such an understanding
* Patients with resected or irradiated brain metastases or those treated with stereotactic radiation therapy are eligible to enroll, provided that they do not require treatment with steroids that exceeds 10 mg of prednisone daily or equivalent
* Sexually active males must use an accepted and effective method of double barrier contraception or abstain from sexual intercourse for the duration of their participation in the study and for 26 weeks after the last dose of study drug
* NaF PET/CT OPTIONAL SUB-STUDY ELIGIBILITY CRITERIA

  * Ability to lie still for imaging
  * Weight =\< 300 lbs (pounds)
  * Metastatic disease confined predominantly to the bones

Exclusion Criteria:

* Any prior chemotherapy or androgen receptor (AR)-directed therapy for CRPC, (e.g. docetaxel, cabazitaxel, mitoxantrone, abiraterone acetate, ketoconazole, or enzalutamide); previous treatment with radium-223 or sipuleucel-T is allowed
* Pure small cell or other variant (non-adenocarcinoma) prostate cancer histology for which treatment with abiraterone would not be considered appropriate
* Patients may not be receiving other therapeutic investigational agents or be receiving concurrent anticancer therapy other than standard androgen deprivation therapy; concurrent treatment with agents to prevent skeletal-related events (such as zoledronic acid or denosumab) will be allowed as long as it was initiated prior to study entry or there are plans to initiate it after cycle 2 day 1.
* Any medical condition for which prednisone (corticosteroid) is contraindicated
* Total bilirubin \> upper limit of normal (ULN) (NOTE: in subjects with Gilbert's syndrome, if total bilirubin is \> ULN, measure direct and indirect bilirubin and if direct bilirubin is within normal range, subject may be eligible)
* Alanine (ALT) or aspartate (AST) aminotransferase \> 1.5 x ULN
* Active infection requiring treatment with antibiotics
* History of adrenal insufficiency or hypoaldosteronism
* Myocardial infarction or arterial thrombotic event within 6 months of randomization, heart failure of New York Heart Association class II or higher, uncontrolled angina, severe uncontrolled ventricular arrhythmia
* External beam radiation therapy within 2 weeks of registration
* Prior history of allergic reactions to G-CSF
* Prior history of allergic reactions to docetaxel and/or to medications formulated with polysorbate 80
* History of active malignancy; patients with a history of cancer that has been adequately treated and are free of disease recurrence for 3 years or more are allowed to participate; patients with non-melanoma skin cancers or carcinoma in situ of the bladder that have been adequately excised are eligible to participate
* Life expectancy of \< 12 months at screening
* Grade \>= 2 neuropathy
* Uncontrolled hypertension (systolic BP ≥ 160 mmHg or diastolic BP ≥ 100 mmHg). Patients with a history of hypertension are allowed to enroll provided blood pressure is controlled with anti-hypertensive treatment