There's a sensor on the furnace (I think called a thermocouple) that senses the gas flow. This thing gets a thin coating of dirt or ash on it and it malfunctions so the furnace continuously cycles on and off. It can be remove and cleaned with emery cloth OR replaced inexpensively. It seems to occur every couple or few years. Its a common problem so googling your furnace make and model should show you where it is. There's probably even a Youtube video.
Have you tried changing the batteries in the thermostat?
Thermocouple senses the FLAME . . . it generates a small voltage that is used to keep the gas valve open. Yes, the thermocouple does get a coating after years of use . . . and yes, can clean it using emery cloth. However, this sounds more like the thermostat itself . . .
If all else fails, call HVAC contractor ....
Good luck.
