The reason you don't want the central air {in the air conditioning mode} or a heat pump in either mode to cycle off and on a lot is due to wear and tear on the compressor motor which is a sealed unit.
I've been out of it for a while but on a house my off the top of my head guess is about $800-$1000 for a 2-3 ton compressor to be replaced. I could be off a few hundred either way but I think that is a low bid. Most problems in electrical motors happen on start up as that is when it is using the highest amperage and under the most stress on the windings.
I saw your next post saying it is gas heat. Cycling off and on too often won't hurt anything in general but the fan motor and relays in that case. It's not yours to worry about though :>}
My guess is you're in a colder climate and your furnace unfortunately for you since you pay the gas bill is trying to heat the great outdoors as well. Here's you a few simple things that don't cost nothing. Daytime winter curtains open. Nights curtains closed as they do have some insulating value. Throw rugs or a towel in front of outside doors to block the air. Keep the filters changed once a month. That is to your benefit. If the air flow is blocked you don't get the right circulation. Sorry I can't help more but if the house is lacking attic insulation {which can be corrected the easiest} it's gonna keep running as heat tends to rise and goes out the ceiling. One good way to kind of tell how good your insulation is serving you is to see how much snow gathers on the roof in say a 3-4 inch snow. If none is laying or it melts off the roof fast within the day you likely have poor insulation.
Thanks again for taking the time to comment.
I've been changing the filter every 3 months, but since you suggested every 1 month I'll start doing that.
If I owned this house, I would definitely be adding more attic insulation. I think that (plus the 1950's windows--no storm windows at all) are the major leak problems here.
Even if I planned on renting for the next 5 years or so, I'd go ahead and pay for the insulation myself. But as I said, I might be moving next year.
Actually, if I owned the house, I'd put in a woodstove heater. To me, that's the best heat around.
I've been able to keep my electric bill in a pretty good range. But the gas bill has been too high, so thought I'd get good advice on FR.
I really do appreciate your replies!