Not a good idea
So I got most of the refrigerator stuff put up. I'm going along my happy little way and forget about the cooler. Many hours later, I discover it. The ice cream inside has melted to a puddle of goo, but is still somewhat edible. The cooler's recently cleaned, so nothing in there's going to kill me, right? I grab a spoon. What's the worse that can happen?
4 hours later, and I still don't want to think about eating again.