I looked up some of their marketing materials. Icy Hot might not be the best analogy. Icy Hot just irritates the skin. It sounds like the heat xfer mechanism might have more in common with sweating. The heat energy of your body causes a reaction in the chemical that produces some menthol smelling gas. Since the reaction requires heat (endothermic), the body loses some heat energy.
When you sweat the water vapor goes thru a phase change into vapor. It’s a highly endothermic reaction so you feel cooler.
Hard to imagine that this chemical can do all that much tho. A driver can burn thru 10lbs of ice in their coolshirt cooler in 40min easy. That’s a helova lot of heat for a heat xfer mechanism that probably isn’t nearly as efficient as sweating.
By the refill kit and soak an old tshirt in it. See what happens.