Love the Mailbox app, but yes, no battery staying power anymore!!! I Googled "does mailbox app drain your battery", because I've experienced such a significant change in my phone's behavior since adding it. I think it's because there's so much swiping and downloading and organizing you do with the app that it naturally wreaks havoc on your battery. Plus you can't fetch email at longer intervals.
I'm one of those people, by the way, who utilizes every tip and trick to retain good battery power throughout the day. I'm keeping the app on my iPad, but have deleted it from my phone!!