David:

Look on your thermostat set up instructions for something called "heat anticipation" or the "heat anticipator" setting.

Every thermostat, even the old Honeywell mercury bulb thermostats had a heat anticipator built into them. Basically, a heat anticipator is nothing more than a small 24 VAC adjustable electric heater that's positioned right beside the bimetallic spring. It's job is to generate heat whenever the thermostat is calling for heat to fool the thermostat into thinking the room is warmer than it actually is, and thereby shut the heat off earlier than it otherwise would.

The reason you need a heat anticipator is many. The most obvious one is that normally the thermostat is centrally located in the heated space, but the radiators or electric heaters are located around the perimeter of the room, usually under windows. So, by the time the temperature in the center of the heated space reaches the thermostat's set temperature, the temperature everywhere else is higher than the set temperature. To avoid this, the thermostat's heat anticipator heats the bimetallic spring to shut the call for heat off earlier than it otherwise would, thereby preventing that overshoot.

Also, different heating systems work differently. With electric heat, as soon as you shut off the power to the electric heaters, that's the end of the heat. With hot water heating, however, once those cast iron baseboard radiators get filled with water at 190 deg. F, they're going to keep convecting warm air into the room for a long time after the thermostat stops calling for heat. Ditto for steam heat; once you get those heavy cast iron radiators hot, they keep convecting warm air into the room regardless of what the thermostat is doing. So, different kinds of heating systems need different heat anticipator settings to minimize the amount of temperature overshoot and thereby provide comfortable heating.

Now, on a standard round Honeywell mercury switch thermostat, the heat anticipator is adjusted by sliding a pointer one way or the other. It's very likely that all that's wrong with your existing thermostat is that it's heat anticipator setting has been turned up way too high so that the anticipator produces way too much heat and thereby shuts the call for heat way too early. Then, if the room is cold, the thermostat cools down quickly and very soon starts calling for heat again, only to have the heat anticipator warm up the bimetallic spring in a minute or two, causing the thermostat to shut off the furnace or boiler again. That is, rapid cycling as you're experiencing.

It is the heat anticipator setting of the thermostat that determines how frequently your furnace or boiler cycles on and off. Too high a heat anticipator setting will result in the anticipator producing too much heat and the resulting rapid cycling of the heating system with the heating system undershooting the thermostat set temperature each time. Too low a heat anticipator setting results in the furnace or boiler running for longer each time it fires and the heating system overshooting the thermostat set temperature.

Too many people have difficulty with thermostats because they don't understand the function of the anticipator. They're constantly turning the heat up and down on the thermostat when really they just need to adjust the anticipator.

I'd just download the information that came with your old thermostat off the manufacturer's web site and find out where the heat anticipator is and how to adjust it. Get on the manufacturer's web site, find out their 1-800 tech support phone number and those guys should be able to guide you to where you can find the original literature that came with your thermostat.