I am aware of the adjustable differential setting which is basically a dead band that the room temp can swing through.

I guess I am not explaining the situation clear enough.

I am looking for a timing adjustment for the on time of the heat pump before the thermostat calls for Aux heat. Say 10 minutes or so. If the outside air is in a range or 40 -35 deg, the compressor will never achieve set point in which case, it will run at almost 100% duty cycle.

It seems logical, in this instance, to ask for Aux heat to more readily reach set point without waiting for the -1 or 2 deg drop in room temp before the thermostat as for Aux heat.

Maybe I am asking for something in a thermostat that is readily not available.

:(

Good luck finding that feature on a thermostat. This has been posted here already today. You will soon find out that the dead band is designed into the thermostat for a reason so trying to find one with out one going to be difficult if not impossible. You have explained the situation exactly and I have done something like you are looking for years ago with a brand called Accustat. Most digital thermostats are adjustable but not to what you want.

See this post

https://www.askmehelpdesk.com/heating-air-conditioning/heat-pump-aux-313484.html

I've read your post and I am not sure if this is of any help. I have a White Rodgers programmable thermostat that will call on aux heat if the temperature doesn't increase in 15 minutes while in heat pump mode. But not adjustable, it's a fixed setting. Unfortunately you can't disable it either, so after a "money saving" setback event, the aux. heat kicks on to eat up the savings. I'm certain there are thermostats that make this an adjustable feature... just haven't looked for one.

Missouri Bound
This can be fixed with a adjustable timer attached to the aux wire before it gets to the air handler. You will have timer activation by the thermostat after the set 15 minute time delay that is built in to the thermostat. This additional timer will allow you to set the activation from right away as it is activated by the 15 minute delay built into the thermostat up to the limit of the time adjustable timer you buy. Been there done that a long time ago.

I have a robertshaw 8425 non programmable tstat and it has a 10 to 40 min delay setting before it will call for the aux heat. I love it. On those cold nights, the unit does not run all night. I have mine set at 20 min to allow the heat pumption to heat the house. It it is still running 20 min later the aux heat comes on and it warms the house right up. Hope this helps