Generally, the APU is used  when boarding to run the air system and other electricals. I think aircraft techs can chime in if the APU alone can power the cooling packs. You may still get air but no cooling.

But from experience on my many flights, the APU only powers the cabin fan. The air conditioning packs are usually powered by ground supply when boarding or after the engines start. If either is not available, you might not get cooling. On a loaded plane during takeoff, they may switch off the cooling packs to maintain maximum thrust and just using the APU to power the fan.

After takeoff the main engines will power the cooling packs and air supply.