Do not put any of those rotten synthetic rugs on in summer. They are only good for when it is really cold and wet, as they do not breathe.
I rug through summer, but to prevent bleaching of the coat and insect bites. Just a very light, UV resistent fly mesh or flag rug is all that is needed. And don't get sucked in by all the fancy colours available, you want a white summer rug, not a pink or blue one. White reflects heat and will keep your horse cooler, particularly if it is a dark coloured horse.