My heat pump is always working, even when the room temperature has reached the set temperature. What is causing this problem?

Possible shorted wire, or a stuck contactor

Is your fan switch set to "on" or "auto"? If its set to "on" it will run all the time, if its set to "auto" it will only run when the system is heating or cooling.

What does auto/on have to do with the heat pump? It's set to auto, but even if it was set to on, only the fan should work, not the heat pump.

So its blowing heat as well as blowing air?

Yes. And I see the heat pump outside running.

A heat pump is made to keep a more constant temp in your house, you may have longer run times in extream cold weather. Unlike its cousin the furnace hot supply temps and heats the house quick and then the house cools off just as quick and then right back on. Question-- is the house keeping the temp you have the thermostat set at? If yes-- its working they way it should, if no-- you have a problem.