A power supply apparatus and a discharge method thereof are provided. A control unit detects a cross-voltage across two ends of an AC safety capacitor to generate a detection voltage, compares the detection voltage with a threshold voltage, counts the number of cycles of the detection voltage during a period when the detection voltage does not cross the threshold voltage according to a comparison result between the detection voltage and the threshold voltage, and performs a discharge operation when the number of cycles is greater than or equal to a predetermined value to discharge electric energy stored in the AC safety capacitor.