To be perfectly honest, I have never really put much thought into cat doors, until a few months ago when I came across this super clever cat door:
I thought this was very clever, and did a search to see what other clever cat doors were out there. Here are some the cat doors that I found:
Cat Door with a brush: http://cathole.com/pages/cat_photos_and_cat_names54.html
So if you are looking to avoid this:
Consider installing one of these super clever cat doors.