Concerts in parks are another big summer thing, and admission to these sorts of things are usually minimal if not free. Pack a picnic basket,grab a blanket, and go get some culture under the summer sky. Besides, if nothing else, you can always go play on the swings. Find out when that is, and go roam around in air-conditioned comfort while looking at pretty things. Get some things to read.
Library cards are free, and they're air-conditioned.
Find some new books that you might not be willing to take a chance on if you were spending money, and be sure to check out the library event schedule. Many local libraries host puppet shows, author readings, and craft classes. For years, my summer wardrobe was a pathetic mix of scruffy shorts, band t-shirts, and the occasional dark floral print sundress. Seems unlikely, but goth style and tropical fashions can peacefully co-exist, even in the same outfit. Tropical prints don't have to loud Hawaiian shirts.