In a lot of games, like LOL and WOW, a used power needs to cool down before it can be used again. Is there a name used for "sending a power to cooldown?"
For the most part, "cooldown" isn't a "place".
It's usually thought of as the property of the ability itself.
Ability does its thing (does damage, or whatever).
Since the Ability has the property "Cooldown, X time", it "goes on cooldown". In other words, it changes it's state to "cooling down" and will spend X amount of time in this state, before changing back to "activatable".
I'm not aware of any separate verb for the transition between the "active" and "cooldown" states. The most common way to express this is to say "goes on cooldown" as I used before.