While the cooler idea is interesting, the moisture might cause the boots to rot and mildew, unless you have a low humidity climate, if left long-term. Perhaps you could use lots of frozen "blue ice" (colder than real ice, and only condenses water from the air, doesn't create its own vapor) in an uncovered bag while driving? Perhaps it's not a big problem for an hour or two at a time. (Caution: I've not tried it!)
It will help to keep your car cooler by putting a sun-shield behind your windshield, crack your windows, or open the sunroof (if you have one) a little. Also - you should have bought a white car with tinted windows - it will be about 15 degrees cooler on hot summer days than dark colors, though that is for our climate. I know one person who had his car painted white, just to keep it cooler - very expensive, if done right.
But: My car is white. I have tinted windows, a sunroof, and sometimes leave it and the windows cracked open, and sometimes use a sun-shield. Near DC (which is fairly hot), it STILL gets hot enough to re-heat-mold skate boots, so they no longer fit right.
Actually, though, my current boots (high end Graf) were meant to heat mold at 260 deg F, rather than the 185 or so that most boots are designed for, and they heat-molded pretty slowly at hair drier temperatures - so I might well be able to leave them in the car... I haven't tried it yet.
Is it a big deal to bring the boots into work with you? It's the obvious solution. I admit you can't do that while shopping. Perhaps you could keep blue ice in a good cooler, and bring it out to put in an open bag with your skates while shopping.