I have a goodman heat pump model #a32-10 controlled by a Honeywell Thermostat model #RTH3100C-installed last winter.
Temperature in South Carolina this week has been ranging in the day in the lower to mid-40's and at night ranging in 20's or teens. I usually keep thermostat set between 66-68. The problem I am having is that once set temp is reached the heat pump continues to run. If I lower the set temp the unit will turn itself off. However, once the new lower set temp is reached the problem with it continuing to run recurs.

A heat pump only does what the thermostat tells it to do unless it is in the defrost mode or has a stuck contactor or relay. If the thermostat is set correctly this should not happen unless the thermostat is having a problem or is somehow wired wrong.

Hw ere is your install guide. Pay close attention to the setup instructions for this thermostat since you have to do the programing to pick the correct system you have etc.

http://customer.honeywell.com/techlit/pdf/PackedLit/69-1889EF.pdf