Oh it's already super hot! lol
If it were Japanese, we go to aquariums, planetarium, movie theaters, shopping malls, etc. How about watching Kabuki performance in Ginza? It will be inside of the cool theater.
Or stay at a temple in Koyasan? It's about 80 minutes from Osaka and since the temple is in the mountain, it seems like it will be a lot cooler in the morning and at night: http://www.vogue.com/article/sacred-mountain-koyasan-japan-shukubo-temple-lodging
This is not exactly an activity but how about trying Japanese shaved ice "Kakigori". Try the matcha flavor one, with red bean paste and shiratama mochi and condensed milk. Kakigori is Japanese summer icon and we enjoy eating it in summer. (Some popular places make long queue!)