This is a life/death issue for my dad who has CHF & cannot tolerate heat/humidity much over 80 without breathing difficulties. We were without power for two days this week (Tues/Wed) after the “derecho” storm came through on Monday. We ran the water pump, lights, etc. on a generator and the temps were cool enough that he was ok. Now, it would be a different story. We have a large generator that will run the air conditioning (heat pump), but for some reason, it won’t start it. Dad is working on that issue now .... might have to go buy a portable air conditioner that will run off the generator to keep one room cool for him if the worst happens. Triple digit temps - tough on any power grid.
It won’t start because the amperage required to start the motor is most likely more than your generator will provide. Starting amps are what matter for equipment like heat pumps, central air, etc. not running amps.
May I also recommend you look into having a transfer switch installed. That way you could plug the gen set directly into your home wiring without having to use extension cords.