Don't work yourself past your limit, take a minute to breath and cool off regularly. Drink a lot of water before during and after being at the barn. Also ride in cool clothes, find breathable pants(I like the kerrits with the grips), and I also really like lululemon shirts, they wick away the sweat and cool you off.
Maybe invest in a fan? Or I often bring those kids juice boxes froze but so when I finished my ride they're very cool and slightly melted, really good idea for shows int he summer! Also how ventilated is your barn? Maybe tack up outside in the shade like near a tree. Also horses feel the heat too so make sure he's getting enough salt/ electrolytes in his feed and drinking enough water