Faced a team in Blitz war, they had 6* ascended Carl. His active skill was maxed at 8. Which says it has an initial cool-down of 2 turns.
I put all 5 of my first round hits on him, then during his first turn he immediately used his active skill to gain AP. There was no command toon on the team.
So, does damaging an opponent charge their active skill? Or have I missed something?