Q. If the AC was left on too low on Shabbos and it bothers the family especially the small children who may catch a cold, can I ask a Goy to turn it off, if I ask him indirectly?
Does it help if I ask him to do it with a shinui such as with his elbow?

A. Igrois Moshe (O.H. 3: 42) permits instructing a Gentile to turn off the A.C. in a shul for the need of people who feel uncomfortable with the excessive cold and will leave the shul.
He also permits to do so at home, even when there are no small children, since in regards to being affected by the cold, all people are considered potentially ill. (O.H. 276: 5.) What degree of coldness is considered harmful is difficult to determine and depends on many different factors (Aruch Hashulcan ibid. 12).
Horav Shlomo Miller’s Shlit’a opinion is similar. He suggests that if the Gentile could turn off the A.C. with a shinui it would be better.

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