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Good question.
It does indeed use more power than push notifications, but the difference should be negligible. My understanding is that Android will batch these events up from all apps, and run them all at the same time.
I've also noticed, on my phone at least, it doesn't bother running the scheduled task at night, and I suddenly get a flurry of notifications a few seconds after I pick up my phone for the first time in the morning. So evidently Android is very aggressively managing when the scheduled tasks actually run.
You can also go into settings have a look at which apps are using the battery, and how much time they spend running in the background. I just looked now, and I'm pleased to see the app is pretty easy on the battery and has clocked "less than a min" of background task time, over the last four hours.
Turning off all notifications does not disable the polling - only logging out does that. And that's a very good point you raise. I'll add it to the list of things to do.
Does that mean more background battery use (app waking up, checking, finding no updates, going back to sleep)? If I turn off all notifications in the app does it stop polling?