Q. Can I do all the above (ask a Goy or place a hot item on the thermostat, see prior questions) if it is too hot and we need to turn on the AC?

A. Turning on an A.C. unit involves more severe prohibitions than turning it off, since it becomes a needed melocho (Tzricho Legufo) and you are actually benefiting from the work that a Gentile performed for you.
However, some Poskim in case of great need are lenient in instructing a Gentile, preferably telling him indirectly to do the work also with a shinui. This is based on the principle of “sh’vus deshvus” or an act that involves two distinct Rabbinical prohibitions performed together, such as telling a Gentile to perform a rabbinical prohibited melocho. The above is permitted when there is great need as when needed for the ill or those in extreme discomfort. (See Shulchan Aruch O.H. 276: 5.) It also maintains that turning on the A.C. is only a rabbinical prohibition (Minchas Yitzchok 3: 23 – 39 Melochos p. 1231.)
Horav Shlomo Miller’s Shlit’a opinion is to be lenient only in severe need. Placing a hot item on the thermostat to induce it to connect, as mentioned before, is permitted in severe need, when a child does it or with a shinui, preferably both.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a