Absolutely not!!
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.