It is recommended to spend as much time indoors during the day in summer to avoid getting heatstroke. However, you still need to ensure that indoor temperatures are cool enough as it is still possible to get heatstroke when indoor temperatures are high!
If you are concerned about electricity bills from using the air-conditioner, how about spending time at indoor places outside of your home? These include cafes, libraries, museums and so on.

Here is a list of indoor places in Osaka City where you can seek shelter from the heat.

Libraries:
Free Wi-Fi spots are available. There are cafes within the facility as well.
・Osaka Prefectural Central Library
(https://www.library.pref.osaka.jp/site/central/english.html)
・Osaka Municipal Library
(https://www.oml.city.osaka.lg.jp/?page_id=8690)
・Osaka Prefectural Nakanoshima Library
(http://www.library.pref.osaka.jp/site/nakato/english-nakato.html)

Museums & Aquariums:
・The National Museum of Art, Osaka
(http://www.nmao.go.jp/en/info/about.html)
・The Osaka Museum of Housing and Living
(https://tinyurl.com/y5e3v5dz)
・Osaka Aquarium Kaiyukan
(https://www.kaiyukan.com/language/eng/hours.html)
・Cup Noodles Museum Osaka Ikeda
(https://www.cupnoodles-museum.jp/en/osaka_ikeda/)

Cafes:
Cafe chains like Starbucks and Doutor Coffee have free WiFi and electric sockets available to charge your electronic devices.
Look up the nearest cafes with charging points on Dengen Cafe (https://dengen-cafe.com/)

Convenience Stores:
Some convenience stores have eat-in sections with seats. WiFi is also available.