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), confirmed =< 4 weeks prior to randomization

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 OR
* 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 that 15 mm to be assessed for change in size OR
* 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 (=< 4 weeks prior to randomization)

Hemoglobin (HgB) >= 9.0 gr/dL (=< 4 weeks prior to randomization)

Platelets >= 100,000/mm^3 (=< 4 weeks prior to randomization)

Creatinine < 2.0 mg/dL (=< 4 weeks prior to randomization)

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 (vasectomy must be combined with a physical barrier method) 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

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, or other immunotherapy-based treatment 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 registration or there are plans to initiate it after cycle 2 day 1

Any medical condition for which prednisone (corticosteroid) is contraindicated

If total bilirubin is > 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) or

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 =< 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 =< 2 weeks of registration

Prior history of allergic reactions to granulocyte colony-stimulating factor (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 blood pressure [BP] >= 100 mmHg). Patients with a history of hypertension are allowed to enroll provided blood pressure is controlled with anti-hypertensive treatment

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

PRIMARY OBJECTIVES:
I. To assess whether the addition of 6 cycles of cabazitaxel to abiraterone acetate in patients with castration-resistant prostate cancer (CRPC) that have previously received docetaxel and androgen deprivation therapy (ADT) for hormone-sensitive prostate cancer (HSPC) can improve progression-free survival (PFS) compared to abiraterone acetate alone.

SECONDARY OBJECTIVES:
I. To assess whether the addition of 6 cycles of cabazitaxel to abiraterone acetate in patients with CRPC that have previously received docetaxel and ADT for HSPC can increase the percentage of change in prostate-specific antigen (PSA) from baseline to week 12 of treatment as well as the maximum decline in PSA that occurs at any point after treatment compared to abiraterone acetate alone.
II. To assess whether the addition of 6 cycles of cabazitaxel to abiraterone acetate in patients with CRPC that have previously received docetaxel for HSPC can prolong time to PSA progression compared to abiraterone acetate alone.
III. To assess whether the addition of 6 cycles of cabazitaxel to abiraterone acetate in patients with CRPC that have previously received docetaxel for HSPC can improve radiographic response (per Response Evaluation Criteria in Solid Tumors [RECIST] 1.1) compared to abiraterone acetate alone.
IV. To assess whether the addition of 6 cycles of cabazitaxel to abiraterone acetate in patients with CRPC that have previously received docetaxel and ADT for HSPC can prolong the overall survival (OS) compared to abiraterone acetate alone.
V. To assess safety and tolerability of the combination of 6 cycles of cabazitaxel and abiraterone acetate.

EXPLORATORY OBJECTIVES:
I. To examine whether patients with circulating tumor cells (CTCs) positive for AR-V7 at baseline have a longer radiographic or clinical PFS to the combination of cabazitaxel and abiraterone acetate vs. abiraterone acetate alone.
II. To examine whether the addition of cabazitaxel to abiraterone acetate can change the AR-V7 status of patients who are positive at study entry.
III. To examine whether the addition of cabazitaxel to abiraterone acetate has any impact on future development of AR-V7 positivity at the time of disease progression.

IMAGING OBJECTIVES:
I. To assess if the changes in total tumor burden from baseline to week 12 as assessed with sodium fluoride (NaF) positron emission tomography (PET)/computed tomography (CT) will differ between two arms.
II. To correlate total tumor burden at the baseline as assessed with NaF PET/CT with the PFS.
III. To correlate heterogeneity of response from baseline to week 12 as assessed with NaF PET/CT with the PFS.

TOBACCO USE OBJECTIVES:
I. To determine the effects of tobacco, operationalized as combustible tobacco (1a), other forms of tobacco (1b), and environmental tobacco exposure (ETS) (1c) on provider-reported cancer-treatment toxicity (adverse events (both clinical and hematologic) and dose modifications).
II. To determine the effects of tobacco on patient-reported physical symptoms and psychological symptoms.
III. To examine quitting behaviors and behavioral counseling/support and cessation medication utilization.
IV. To explore the effect of tobacco use and exposure on treatment duration, relative dose intensity, and therapeutic benefit

OUTLINE: Patients are randomized to 1 of 2 arms.

ARM A: Patients receive abiraterone acetate orally (PO) once daily (QD) on days 1-21, prednisone PO twice daily (BID) on days 1-21. Courses of abiraterone acetate and prednisone repeat every 21 days in the absence of disease progression or unacceptable toxicity. Patients receive cabazitaxel intravenously (IV) over 1 hour on day 1, and treatment with cabazitaxel repeats every 21 days for up to 6 courses in the absence of disease progression or unacceptable toxicity. Patients also receive standard of care antiandrogen therapy with either luteinizing hormone-releasing hormone (LHRH) agonist or antagonist, or surgical castration with bilateral orchiectomy.

ARM B: Patients receive abiraterone acetate and prednisone as in Arm A. Patients also receive standard of care antiandrogen therapy with either LHRH agonist or antagonist, or surgical castration with bilateral orchiectomy.

After completion of study treatment, patients are followed up every 3 or 6 months and then annually for up to 5 years.