In a 30-day campaign, 25,000 AP (on a single character in a single campaign) is the magic number for getting you a guaranted 50-transmutation-crystal geode.
Note that you can do this multiple times per character per event, in effect, by refreshing the AP buff before the event ends, then not playing the character for significant time until the current campaigns and new ones start (July 18, approximately).
But remember to set a home campaign!!
The standard way to get easy AP is to repair walls. After a keep gets new ownership, it periodically increases the max health of its walls. This yields a lot of opportunity to repair walls at various places in the keep. Surely there are more detailed guides to wall repair people can link.
I really started this thread to share another tip. If the keep closest to one of your alliance's home towns is captured by the enemy, they probably won't hold it for long. Participating in its recapture can yield a bunch of AP. And the "participation" can be as minor as just showing up or, to be on the safe side, attacking a few NPC guards after your allies break down a door.
Scouting quests can of course give a small amount of AP (1000ish during the event) while requiring no combat at all. Either just do them while evading combat, or log out until such time your alliance controls the keep you're supposed to scout near.
Even simpler and more profitable, but less reliable, is to accept the Elder Scroll quest current available, and hope other people take care of the goal while you're still logged into that campaign. If they do, you'll get a quest completion worth a few thousand AP.