I uh, wow...
Update:
This does not seem to effect other modes:
This thread on GameFAQs implies it was originally 20 minutes in the campaign as well: http://www.gamefaqs.com/boards/676176-jojos-bizarre-adventure-all-star-battle/67175142I don't, but it's related to the game's Campaign Mode.
Basically you have a battery with 10 bars that refills one bar every 5 minutes while in Campaign Mode or every 20 minutes outside of it. You spend one bare for every fight you play, and there's bosses to beat that require multiple fights and/or using more bars of energy so they take more damage. There's also random events, some that use energy too.
This mode needs to be played to unlock alt costumes/colors/quotes/poses and the bosses hold most if them.
You can buy battery refills and consumable special items to find bosses more easily through the PSN.
Update:
This does not seem to effect other modes:
No. The online mode is just the usual fighting game lobbies. The campaign mode is a special errr... "story mode" I guess. Except instead of story mode it uses the online server to randomly generate events and stuff which saps your energy bars. It's really like a social mobile game in design, complete with leaderboards and stuff. But it's not multiplayer.
It's completely separate The (crummy) story mode, versus, online, yaddayadda all have no limits whatsoever. This is kind of like the gashapon mini-games in Smash Bros, only the rewards are customization stuff, which is more relevant to more people.