Suddenly, my heat pump comes on for about 10-15 seconds and then turns off. This cycle continues until I switch the thermostat to off. Do I need a new thermostat or is the problem related to the heat pump itself?

Its is hard to say... we would need more info. Like fans running, compressor running, how long has this happened, did anything happen that could have changed the system, electrical storms?
The more info we have the better we can help you out.