If you have a classic single-stage air conditioner, not much. Your thermostat just controls whether or not the AC is on or off. If it’s too much above the desired temperature, it goes on, and once you get down to the desired temperature, it turns off. It doesn’t actually output differently depending on the desired temperature, it just stops outputting when it reaches the target temperature.
But there’s also something called a two-stage air conditioner. Basically, it can run the actual air conditioner at reduced power in a more efficient mode. It will do this if it’s closer to the target temperature. If it’s further away from the target temperature, it runs at full power to get cooler faster.
So if you make a sudden change to the desired temperature on your thermostat, on a single-stage AC it can trigger it to turn on, but if it’s already on it just changes the threshold for it to turn off. For a dual-stage AC, you could also cross the threshold to change it from high-efficiency to full power mode.
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