TamaDroid GAME
This app runs an optional background service, so battery life is only affected if the service is enabled. Without the service, closing the app shuts it down like any other. The pet's current state on shutdown is saved, and when the app is reopened it calculates what the pet did in the meantime. You can turn off your phone and use it as usual of course. By accessing the settings menu you can choose when the pet sleeps and wakes up, the desired background image and whether evolutions should be random, among other things. You can also pause the pet, if you need some time off.
*To those complaining the app has frozen: Your pet has died when it either becomes a tombstone or an angel, and you can no longer feed or play with it. At this point you should reset it.
*Translations are currently available in English, Spanish, French, German, Italian and Norwegian. If you want to help translate the app into your native language, contact me.