In everyday English, you can say turn up the heat or AC. You can also say " turn down or lower the heat" because the house is cooking. If you ask someone to turn down the AC, you are saying it is too cold.
But I hope i'm not being rude by asking if you are absolutely sure that it is expressed this way through out US, because just recently someone (a native speaker) told me she meant for me to "dial down" the temperature on the AC control when she asked "could you turn down the AC?", to which I had her clarify because the room was warm!. That's why I asked.