Currently I use Pocket Casts. The version 4 update makes it a very good app, and it's the only podcatcher that I know of to have built-in support for variable speed playback (1.5X, 2X speed) of audio (i.e. that doesn't use the clunky, buggy, separate purchase, forever-in-beta "Presto" library). Note: now there are alternative libraries to Presto, including the free open-source Prestissimo.
Unfortunately it still doesn't support variable speed playback for video. The iOS podcast apps have had this for a long time, so it seems that something must be built in to iOS. There do seem to be Android apps that support variable speed video playback, but no Podcatchers.
Jason Howell (from AAA) posted about this not too long ago on Google+ and got nothing: https://plus.google.com/115109077044202722504/posts/MfdG5gF8BcW
Could watch for updates at:
AntennaPod seems like a decent podcast client, and it's open-source (https://github.com/danieloeh/AntennaPod). They've been working on variable speed audio (https://github.com/danieloeh/AntennaPod/pull/184).
Since I do mostly watch video podcasts at work and at home, I could just use a PC (the gym might be one exception).
Look at https://gpodder.net/ (sync service), which may also work with some Android apps.
Some apps to try:
To look for / to try: