That's a weird situation. Anyway, if the router changes it's IP-address, it needs to broadcast that change to the world somehow; if it wouldn't, packets would not be able to reach it. Before implementing any kludge way, I would first find out what this router is exactly doing, and why. Once you know that, you might be able to use a more reliable solution.

Note that adminning a box through a router that changes its IP address on a regular basis is at best tricky, and you might find yourself locked out. When the router changes IP address, all your open connection are suddenly stale.

Abigail