I've tried all sorts of stuff like camp-running, combat training, fishing etc but this is by far the best I know of. Using this you'll get anywhere between 60-100+ AP every 10 mins, depending on your luck. TLR at bottom.

Very shortly into chapter 4, you'll gain an item (as part of storyline) that allows you to summon a random number of enemies.

Go to one of the desert areas around Hammerhead (I like going south- ish), preferably near some rocks to which you can warp-point to replenish mana.

Use the item, which should summon either Lvl 2 Sabertusks or Lvl 1 Scorpions (usually between 2-5 of them) .

Equip a high attack weapon (a greatsword works well), and warp-strike to kill each enemy, which will grant you 1AP per kill. Repeat using item as much as desired.

(Optional #1) Use warp-points to replenish mana for lower downtime.

(Optional #2) Once you get powerful enough/high level and a really good weapon, you could even 1-shot the imperial soldiers which come off the ships. This works insanely well because there's lots of them all clumped up right as they disembark.

(Optional #3) To reduce downtime EVEN more, try to use the item again the moment you kill the last enemy in the pack. This reduces the lockout period where you can't re-use the item while exp is being tallied.

TLR - Get to Chapter 4, use item from story to summon weak monsters, warp-strike to kill for 1AP/kill.