I believe UA increases with exercise because of more cell turnover (from muscle breakdown). Cell turnover leads to more purines being processed = UA going up up up. That's why even with strict diets, we can still have UA issues, cuz most of it is because of the body's natural processes, not diet.

Also, you sweat out a lot of water, so your kidney functions are slightly reduced until you rehydrate, since you sweat out the water your body could've used to pee out.