Whoever told you that something is making the battery drain fast is dead wrong: If the battery were drained, the motor would not start later on; unless the battery was first recharged. Your problem most likely comes from a defective starter solenoid, or a defective starter relay. Since it is heat related, I would think it more likely that it is the solenoid; because the solenoid is mounted on the starter, which soaks up heat from the motor.
You know; considering that it starts when the battery is jumped; this sounds like corrosion has gotten into the battery cable contacts. Corrosion of this type is often not visible to the eye. An easy test to see if this is causing the problem is to turn the headlights on before trying to start when the motor is warm, and watch their brightness when you try to start the motor. If the lights dim way down or go out when you try to start, the battery terminals and cable clamps should be thoroughly cleaned; using a tapered reamer type battery cable service tool. If the lights do not dim; the problem is in the solenoid or starter relay.