Shopping (somewhere where there’s central air), local pool/lake/river and head to an air conditioned bar (if you’re old enough to drink), or check out a museum (if you have one around you).
When I was in Austin with my ex. (she was from there) it was a lovely 108 degrees. We went from her air conditioned apartment, hit a few stores, hung out at Barton Springs, then ended up hanging out at Manuel’s and drank margaritas all afternoon. We went home and showered, then went out dancing. We managed to stay cool.
While I was there, though, we went to Austin City Limits, which was outdoors and it was hotter than hades. I would often pour a whole bottle of ice, cold water over myself, drank plenty of cool beverages and chilled under shady trees.