Have carrier furnace with Honeywell programmable stat. hen in heat mode unit keeps running and when the stat reads setpoint it stops for a brief second and goes through the start up cycle again. This continues until the outside temp raises to 40 deg or higher.What could be the problem.

Is the honeywell tstat a new install or has it been there a while?

About 1 year and the problem existed with the old Honeywell T8600D Chronotherm IV

Have you considered the fact that this has done the same thing with two different thermostats and the possibility exists you have a large heat loss in your home calling for the thermostat to run the furnace again right after it turned off? The possible give away on the problem is the fact that after it warms up a bit to 40 degrees or so the demand that the furnace run more often is negated by the fact that the demand for heat is not as great as it is at a colder temperature.

Have you checked to make sure the hole in the wall behind the thermostat where the wire comes out has been caulked or plugged tight to prevent possible cold wall air from causing thermostat problems?

When you ran the setup for the thermostat did you make the correct selection for the type of furnace you have?

What is the exact model number of your thermostat so we can look it up to see what adjustments are available?

It is possible it is a furnace problem BUT it seems more of a thermostat problem to me at this point with the outside temperature playing a part in the problem.