I raised rats all summer in southern California! I have a little room between our garage and a cinder block wall. Doesn't get much direct sun, that's the real killer. I ran a fan off of a timer. You might be better off with a leanto in the shade than a hay loft. If you have space in your tack room, maybe you could keep some there. I know we would always catch mice if we left the top off the sweet feed. They'd fall in the garbage can.
Remember though, even when it's 80 or 90 out, there may be some place where it's cool, damp and the soil is not being baked by the sun.
Good Luck and keep us posted. Your smart to think about this stuff first.