Android works in a way where apps do get saved into memory so they can get retrieved faster. Especially stuff like Amazon MP3 benefit from it, otherwise it'd have to cache your thousands of music files (well mine anyway, ha!) when you open the app.
Don't worry too much about the heating up. It has more to do with the kernel's CPU governor and clock speed. You can't edit those on the Amazon ROM though.
If you think you have a battery issue, a lot of people here use GSam.