My bug report was closed, as it was a duplicate- so there are others out there reporting the problem. Also, the reviews on iTunes are...unkind to say the least.
I noticed a battery drain on my iPhone since getting this app (or possibly iOS 6, not sure about the exact timing or cause).
I manually turned off the sync for each podcast, and it seemed to have helped.
I also manually closed the app.
I just turned on the auto-download for each podcast; and then turned off the app after 5 minutes. So if this helps, it might be a decent work around.
The last thing I am trying is I set my iTunes to check daily, rather than hourly for podcasts. Since the app lacks any controls, I am wondering if it is getting at least some data from iTunes on the computer.
Closing the app when done might also help with it turning itself on when you hang up.
My podcast app also has trouble with keeping the number of episodes per podcast right. I am currently trying to keep the # set to keep on iTunes the same as the app. So far, mixed results on that.