Generator for AC

Yep, it should. Like John&Kris, I have the Honda 2000i.
The general take is you need that 2000w for startup but once running not near that. When I start the air it doesn't even throw the Honda into high gear. I wouldn't be surprised if you could run the AC with a 1000 watt'r but they say you can burn out the compressor if too low.

boo - If I wasn't out boondocking (photography) so much, was just a normal camper, I'd probably not have invested in a generator and just head for campgrounds with hookups if I needed air. A 2000 watt'r isn't the lightest thing in the world, takes up space and you have to carry gas. To run the air around the clock you'd need 3-4 gallons of gas per day. My main use is to keep my computer charged while photo editing and for rainy days when solar won't work.

My only caution would be that it may not work at high altitudes.  My 2000w Champion generator would not run an ac at 6600 feet (Signal Mountain Campground in Grand Teton NP).