Hey, Skie Fortress, thanks for trying the old demo.
The AP regeration is constant even if you have leveled up. If you want to get the AP faster, defend. It could adds 7 AP and for each turn, you will get 5 AP. So, you get the profit by defending. Also, if you're lucky enough, there's accessories could boost your AP regen. Or if you don't mind, use Action Regen item.
And of course, since it was greatly inspired from E:D! there will be timed-hit soon in later version of the battle system. I just need the time