Are you using a hydration pack? I would think that tucking it in next to the bladder would keep it cool enough. Tuck a small thermometer (like those keychain dealy-bobs) and check the temp. Maybe wrap it with a piece of neoprene.

If you can spare the bottle cage (yeah, in Arizona ), maybe a cage rocket or whatever those bottle-shaped boxes are (I think TE has them) with a partially-chilled gel pack would work.

I don't have the problem but maybe these are some ideas.