I have a Dell router, which overheats a fair amount (Once every week or so). It's mainly because we run a Teampeak server 24/7 off of my brother old Dell Dimension 4400. We needed a server, so we turned his old machine into one when he built his own PC.
lol that will probably be ok because there will always be COLD air available so it should be ok.
but interesting question: will the sweat from the cold pack warming up drip down into the router, get some metal salts from the dust inside there, and short it out? (remember, pure water doesn't conduct)