quora answer, but sounds knowledgeable:
You have three power sources on an airplane: the engines themselves that have an in-built generator, an additional, small engine that’s in the back and just for electricity, that’s called the APU or Auxiliary Power Unit and third, while on the ground, there’s the GPU or Ground Power Unit which is essentially a power cord that’s plugged into the plane.
AC can run off all of these power sources, but:
The engines are usually not running while boarding for safety reasons
The GPU is rarely used. Also many airports don’t have them for all airplane models.
So essentially, that leaves us just with the APU.
The APU however, on hot days, often can’t provide enough power to effectively run the AC, which is why AC is often turned off on the ground to keep the power for other appliances.
But apart from all that, and that’s the real reason for many flights: the APU is VERY often broken. And until the routine checks at it’s home airport, it won’t get fixed as it’s not directly important for safety.
And well that’s why the AC often isn’t working before the main engines are turned on …
Have a great day!
EDIT: As Marius pointed out correctly you’d have a problem if the engines are off, the APU is broken and there’s no GPU. You wouldn’t have electricity for the Cockpit etc. Fortunately, however, every airplane has batteries capable of providing the necessary appliances with electricity! And of course, if an airline knows their APU is broken, they will be doing everything possible to get a GPU.
https://www.quora.com/Why-doesnt-air-conditioning-seem-to-work-when-a-plane-is-on-the-ground-but-works-perfectly-in-flight
Might be OK for ordinary temperatures, so here the problem was deciding that was OK on a really really hot day, whether technically allowed or not.
= new reply since forum marked as read