"... in my experience that cookie continues to work even if you log out..."

Honest question: Should logging out invalidate the cookie for future use. If one of these cookies falls into the wrong hands it grants "evergreen" access to the account. No password needed. I have noticed this behaviour with both Gmail and Facebook. These evergreen cookies makes it very easy to check FB and mail using shell scripts without logging in, which is nice, but it seems like it should be a security issue.