Apalutamide Plus Intermittent Hormone Therapy Versus Intermittent Hormone Therapy Alone in Prostate Cancer

This study is open to men who have biochemical recurrence (BCR, increased PSA) following local treatment of their prostate cancer. Androgen deprivation therapy (ADT) is a standard treatment option, but is only effective for 16-24 months and has a number of side effects that impact quality of life. These side effects may include fatigue, hot flushing, loss of sex drive, brain fog, decreased bone mineral density, loss of muscle mass, mild anemia (low levels of red blood cells that can make people feel tired and weak), diabetes (low blood sugar), heart disease, metabolic syndromes (sometimes called "pre-diabetes" and includes obesity, increased blood pressure, high levels of cholesterol and triglycerides in blood), and risk of fractures. An alternative to continuous ADT is intermittent administration, where patients are given "breaks" from ADT to let their testosterone levels return to baseline. There are a number of potential benefits to intermittent hormone therapy (IHT): (1) longer time to the development of resistance; (2) improved patient quality of life owing to recovery from adverse effects, particularly sexual function; and (3) substantial cost savings owing to less time spent receiving medication. Leuprolide is the name of the ADT / IHT drug. Apalutamide is an investigational drug, which means it has not been approved by the Food and Drug Administration (FDA). It is an antitumor drug, taken by mouth. The purpose of this study is to determine the ability of Apalutamide to extend the time between the first two injections of leuprolide and improve quality of life. This study will also look at the safety of Apalutamide and the effects that Apalutamide has on prostate cancer. Men will be randomized (like flipping a coin) to receive: * Group A: Leuprolide + Apalutamide or * Group B: Leuprolide only (until second leuprolide injection), then leuprolide + Apalutamide 45 men will be in Group A and 21 men will be in Group B. Leuprolide is given as an intramuscular sho