Facebook’s Graph API doesn’t have a method to easily retrieve a cover photo. However, provided that you (or a user) has given your app access to their Photo Albums, you can get the photo pretty easily.
The best way to get it through a single request is using the following FQL query.
SELECT cover_object_id from album where aid IN (SELECT aid, name FROM album WHERE owner=me() AND name="Cover Photos")
Note that you will need an access token to use this query. It definitely works for me, but I’m still testing with permissions to see what’s needed to make it work with other users. Will post an update if anything changes.
Update: Yep, this works. Have it running on my iPhone right now. Remember to change “me()” with the Facebook ID of the user. “me()” will only work from the FQL console.