Heat pump thermostat with small differential

Gee pretty hard to do when the application is unknown.

I don't know specific model for a heat pump, but there are Honeywell electronic thermostats that learn their anticipation (comparing rate of change of air and wall temperature) and usually hold the temperature fairly evenly (model CT3500 usually varies less than 1 degree F even for my steam heat). It can be set for different number of cycles/hr depending upon type of heat. It only overshoots slightly temporarily if we get sudden mild weather following cold weather.

My previous thermostat (another brand) had minimum +-1F setting and no anticipator. It would drop -1F below setting, fire to +1F above setting, and continue to climb due to hot radiators (3-4F swing).