If my house is not so cool. Can I have a goy come turn the air lower to make my house cooler on Shabbat?
The goy should not come to lower your a/c for you on Shabbos. the reason is because essentially we may not ask a goy to do melacha for us on Shabbos, in fact if he does do melacha for us on Shabbos even if we didn’t ask him to do it, we may not benefit from that melacha until after Shabbos. There are some exceptions though, for example it is permitted to ask a goy to do for a sick person. The poskim discuss what about asking a gentile to turn the a/c on if it is very hot outside, is it possible to say that if we are very hot that it is as if we are sick… Regarding your scenario, if the a/c is on and you merely want it lowered, it is hard to call this , that the people are sick, in order to necessitate having the goy lower the temperature, which involves melacha. Therefore it should not be done on Shabbos.