Weatherproofing the cat door wasn't on my original to do list, but it's been cold here in Austin and I realized just how much we needed to. The entire study felt like we were running the AC. It was probably 5-10 degrees cooler than the rest of the house.
The last owner put the cat door in and left some pretty big gaps between it and the rest of the door. But the addition of some weather stripping and expandable insulating foam have sealed things up nicely; we're no longer heating our backyard through the cat door.