I use Facebook for an author page. I don't know that I would make a page specifically for a book. Mostly because every time I write something new, it would need a new Facebook page. For me, it is easier to keep everything in one place.

I have seen authors make fan pages for characters, but these are usually series with a big following. Sherrilyn Kenyon is famous for this. She had a page for Acheron, who is her best known character.