There are lots of misconceptions, so I just want to clear it up. When is one permitted to ask a Gentile to turn on/turn off/adjust the air conditioning on Shabbos?
My understanding (I don't have sources handy) is as follows:
The Gemara says that if people are cold, we treat them as "seriously sick" and therefore allow asking a non-Jew to heat up the room for them.
If people were hot, back in the day there wasn't much that asking a non-Jew would help. But today, R' Moshe Feinstein allowed asking a non-Jew to turn on the air conditioning if people are too hot.
(If people are seriously too cold, you could even ask a non-Jew to turn on the furnace, so I don't see why he couldn't turn off the A/C instead. Or adjust it, if it's too high or low.)
Of course, check with your rabbi, but for some kinds of thermostats, it is possible even for a Jew to turn up/down the AC on shabbos, as long as the action prolongs the current state.
e.g., if you are too hot and the AC is not currently running, wait until it gets a little hotter, kicks on, then turn down the thermostat, thereby prolonging the cooling stage.
I have seen some people use a shabbos clock with lots and lots of switches on/off/on/off/on/off... so that if they want to change the temp, simply wait until the AC is the way you want it, then flip then next few switches all to the same.