Is it cheaper to leave the air conditioning on or turn it off?
The physics behind a genuinely contested question, and the conditions under which each answer is right.
Reviewed
The short answer
Turning it off is cheaper, in almost every case. The widespread belief that a house costs more to cool back down than to hold cool is, in general, false.
The reason is that heat flows into a house in proportion to the temperature difference between inside and outside. A warmer house leaks in less heat. Every hour the system is off, the house is gaining heat more slowly than it would have if held cold.
Where the myth comes from
Recovery is visible; holding steady is not. Coming home to a hot house and hearing the system run flat out for an hour feels expensive, while the same system quietly cycling all day does not.
But the recovery run is finite, and the cycling you did not pay for while you were out was continuous. The arithmetic almost always favours the house that was allowed to warm up.
The real exceptions
Humidity. In humid climates a system that has been off must remove moisture as well as heat on recovery, and comfort may not return as quickly as temperature. A setback of a few degrees rather than switching off entirely handles this.
Heat pumps in heating mode. Allowing a house to fall far below the set point can trigger backup resistance heat on recovery, which is expensive enough to erase the saving. Keep setbacks modest.
Anything with a hard temperature requirement — pets, plants, instruments, medication.
Very short absences. If you are out for an hour, there is little to gain either way.
What to do instead of arguing about it
Use a setback rather than a switch-off: a few degrees warmer while out in summer, a few degrees cooler while out in winter. This captures most of the saving and avoids every exception above.
A programmable or smart thermostat does this reliably, which matters more than the size of the setback, because the saving only exists if it actually happens every day.