Well, here in Tucson, Az, where the temp is generally over 100 degrees by 10am, and stays such well past 9pm, what we do to stay cool? We sweat. Pretty simple!
Not unusual to see folks just doin' their thing while covered head to toe in a nice lather of sweat! Now, for the horse's needs, basically, it's trying to ride as EARLY or LATE in the day as possible, so at least that DIRECT sunlight isn't an issue for them. We ensure a constant supply of fresh, tepid, (not cold) water for them, and LOTS of cooling down prior to returning to the stall...I always feel between the breastbone area...if that is moderately warm or damp, even in the slightest, (regardless if the rest of the horse is seeming cooled out) then they are not nearly cool enough.
Also we don't ride really hard during the high sun times. It is just not possible to keep your horse as cool as you can keep yourself internally, as they tend to not drink so readily during an arena ride!
We will wait until their inner temp (judged by chest feel) is pretty close to water temp (tepid water) and then do a nice hose down in a covered wash rack so the wet parts don't reflect the sunlight and heat them back up again like tin foil in an oven! But sadly, unless you can ride and be done by 7:30a or after 8p, you're going to be VERY hot and guzzling water like there is no tomorrow and your horse is going to be pretty miserable as well.
It's funny, because when I read about the lack of winter riding that some folks are able to do because of the insanely cold weather where they live,(and they always say how much they can't wait for summer to begin really training!) I always think that it is pretty much the OPPOSITE here in Tucson! We live for winter, late fall, & early spring riding, whereas summer riding can be very tricky!
Good luck with staying cool! (My barn has a completely indoor, air conditioned arena, but it is an odd, long shape with very short ends. It is doable, but only as a last resort--like, I would imagine, on mid-summer days that start out at 90 degrees by first light and don't dip below that ever!)...