Our Gas furnas does not sense that the temperature is getting lower than the set temperature. We thought it's a termostat and we bought and installed the new one and still it does not function. Of course, when we turn the heat on and off and increase the set temperature, it does turn on and works for a while but after we don't touch it for an hour or so, the sensor does not work and temperature goes down until we turn the heat off and then on and then the sensor starts to function.

Many (most?) gas furnaces have a small window in the blower compartment door through which you might be able to see a blinking light. The next time the furnace shuts down, before you turn the stat down or off, count the blinks then open the blower compartment door & look for a sticker with a key to the blink code. Let us know the meaning of the code & we can probably help.