The Loot Hunt buffed the effect to about 25% per kill according to my tests.
So while your Action Skill is on cooldown, and only then, every kill with the Evie or anything else you cause while holding it increases the rate by 25% each. This is the rate we are talking about here so if you were to kill 12 enemies you’d increase the rate by 300% resulting in your AS cooling down four times as fast.
As Gaige, I killed exactly 12 enemies (hooray for test dummies) and the 60 seconds became 15 almost exactly which is how I got to my number. This also appears to be retroactive so that the new rate saves you the seconds as if it had that rate since the AS started cooling down.
That rate is reset once you’re able to use AS again of course.
I made so many mistakes testing this, sorry for any misinformation on my previous edits >_<