How to prevent my cutting mat from warping?
Taking each aspect in turn:

Covering the mat with something white, a sheet of paper, a fabric sheet, a light colored towel, may provide protection from any thermal related warping. If the mat is warping in a manner that causes it to lift upward in the center, generally speaking, you may be getting "top layer" thermal expansion.

Hammering is less definitive, unless you can see patterns in the mat. If you are hammering in a regularly visited location, it may be splaying out under the mallet or hammer, causing a sort of bottom layer expansion, which would result in the edges curling upward. It would take substantial hammering in the same location to accomplish this.

Small sections of localized bending could be the result of hammering. It would result in an irregular form of warping, perhaps dimples in some areas, bumps or moguls in others.

I found this on the Blick web site:
"If your Blick Cutting Mat should become bent or distorted, simply heat it in sunlight or hot water until it becomes pliable and then lay it flat until cool."

You could consider to play a hair dryer over the surface to "relax" the warping or bumps. I would recommend against anything as powerful as a heat gun, as it has the capabilities to melt the surface in a very short time.

If you are successful in flattening the mat, shade and cooling air flow are your friends.