Turning off the generator with the AC going?

What he said above ==

no problem with turning off the gen under load. No different then if it ran out of fuel. A/C s not on the inverter. Gen will to start an A/C until it is up to speed. Built-in delay.

So far as the generator is concerned, the AC thermostat "on" but not actually cooling is no different than the AC thermostat "off." In both cases, the air conditioner is not running and hence not consuming power.

As was said, best practice is to turn off loads before shutting down the generator. However, it's generally unlikely that failing to do so accidentally will cause trouble. It's in the category of things to avoid, but not in the category of things to keep you up all night worrying about should they happen.

Shutting the thermostats off before starting the generator is strongly recommended; starting under load, especially a heavy load such as two air conditioners starting up simultaneously, is rough on the generator. Let it start up and stabilize (and ideally warm up a bit), then add the significant loads individually so far as it is practical to do so.

rk911 wrote:
our AC does not run off the inverter. yours is likely the same.

This. It would have to be a honking big inverter to run AC.

our AC does not run off the inverter. yours is likely the same.

Best practice to start and stop a generator with minimum load. Hard on the regulator and engine to start and go immediately to high throttle. Hard on the regulator trying to maintain voltage as the engine winds down. 30 seconds is generally enough wait time.

“Does it hurt to turn the generator off with the AC's running?”

Read your manual. My generator booklet says to remove loads and run for a short time before turning off.