The indicator isn't acting weird. It is designed to come on when the circuit with a bad bulb is energized, and not before, and it resets itself after turning off the ignition.
It is telling you something is amiss in the brake light circuit. Check out JCE's tip above. You should at a minimum remove and replace the brake lamp bulbs, to clean the contacts, and don't forget the 2 bulbs in the third light inside the car.