I got to work on these windows a decade ago! They use basically the same chemistry as in auto-dimming rearview mirrors.

Feel your mirror sometime when you're driving in traffic at night on the interstate and it's been dimmed because of the vehicle following you for some time...it will be perfectly cold.

The electrochromic material between the panes goes through an oxidation/redox reaction in the presence of a voltage. In circuit, it functions something like a low-storage battery or capacitor. It does not have enough leakage current to cause it to get hot, that's just the effect of the sun on the plane.