I've always found that in situations like this one, it works well to identify with them. Remind them that it is hot for all of us and we will get it cooled ASAP. Had a couple times at XJT with MEL'd APUs I coordinated an early engine start. If we could do turns with it running then no reason we couldn't crank it up early for some air. I've also refused to board if it was too hot and magically a working air cart was found.