I'd always wondered if there were a such thing as a propane-fueled refrigerator. Was even discussing the concept with my wife a couple of days ago.
In my theory of how such a device would operate, basically, the burning part and the cooling part are not related. The burning part is only responsible for turning the shaft of the compressor pump. The heat from the propane burn is exhausted in such a way as to be separate from the heat exchange mechanism of the fridge itself.
Dunno if they actually work that way, but seems logical from the way I was thinking about it.