First off, without getting into the real science (most of which is beyond my knowledge), it's "generally" considered bad practice (for the health of the water within) to keep a rain water barrel in full sun.
Coming to the question, your temperature is definitely hot enough to considerably warm up the water inside a rain barrel exposed to full (summer) sun. Personally I wouldn't use warm water (if cooler water was an available option) on vegetable, shrubs or other plants in the garden. But if I had no other choice, then of course I would use it (via watering at soil level, not directly on the foliage).
I was always told to use cold water when watering plants etc. I never really questioned "Why?". I have read about using (and have used) "tepid" water (as per Mancuniensis' answer to this question). During the heat of the summer here in Missouri (think ridiculously hot) I always discharge the hot water that has built up in my outdoor hose before watering the garden (with cool water).
If you cannot move your rain barrel into the shade, I would suggest something like filling watering cans, buckets, etc. from the rain barrel, then putting them in a shady area for a day or two before using that (cooler) water in the garden.