If overheating is a concern, I would stay away from using cardboard boxes to insulate it. I'd be worried about the fire hazard when using paper products that close to a pump.
Using the foam material that is shaped like an egg carton on the interior of the closet should help deaden the sound. There's also some specialty stuff sold by audio suppliers that works. (Such as the stuff put underneath cars to deaden road noise). I would imagine that it wouldn't be too cost effective though...
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