In theory, you could cool the icebox or its contents with a system of ventilation ducts, possibly with fans, that open at a specific time or temperature. In practice, it will be hard to make something reliable - particular if you want to make a hole in the ice box. Hard to do that and still make it an efficient insulator in the day time...


The piezoelectric coolers is a nifty idea, but their capacity is too feeble. They can keep the box at up to 20C/40F colder than its surroundings, but will use like 50 watts to do so. If you run it for a limited time (such as for a designated interval, or you could just run it when the engine is running) that time interval is probably too short to make that much of a difference.


The most efficient would probably be to to run that cooler box system for a short while in the middle of the night, replacing that air within the box with cool outside air that is further piezo-cooled just a little. I think you could make a measurable difference with a relatively small investment of the power of your car battery. If that difference is worth the effort remains to be verified.