There IS a more modern, concrete hotel just down the road, which has AC. (The Green Hotel doesn't have AC). I was tempted, because even in September it was pretty hot in South India, and in a lot of the smaller places we visited, everything was concrete, basic, even shoddy, and an AC unit was a major selling point.
But the Green Hotel is a lovely period building, once a summer palace, and is naturally cool and shady inside. It's also the only "eco-friendly" place in town, hence the name. We loved the place - it was a much-appreciated oasis of peace, cool and comfort. Ignore the "luxury" hotel down the road and stay here if you're ever in Mysore!