What's the AP's POV?
The AP's claim is based on: (1) copyright law -- you cannot rewrite the article in a manner that creates an infringing derivative work, (2) state misappropriation law -- there's 90-year old case involving the AP that says you can't compete unfairly by stealing "hot news," and (3) wishful thinking -- copyright permits free use of facts.
Beyond Copyright
There is a legal theory that goes beyond copyright which prohibits the theft of "hot news." It's based on the 1918 case of International News Service v. Associated Press in which a rival news agency 'stole' and 'rewrote' AP news articles. Some states still uphold this approach; others consider it invalid. (The AP settled its most recent hot news case) It's unlikely that the "hot news" principle will apply in your case because the publication of your book would not trigger the "time-sensitive" aspect of the hot news doctrine.
Just the Facts
Aside from the "hot news" doctrine, the AP cannot stop you from taking the facts of an article and writing a new article using those facts. Consider this recent AP article about the death of Les Paul. You can certainly "rewrite" the first few paragraphs in your book as follows without infringing."Les Paul, 94, credited by many as the inventor of the solid-body electric guitar, died Thursday at White Plains Hospital from complications from pneumonia. Paul is also credited as one of the pioneers of multitrack recording, a process that permits musicians to record different parts at different times and then mix the tracks together." (Copyright aside, we urge every musician with a home studio to observe a few a moments of silence for the man who created modern recording!) Finally, the Dear Rich Staff wants to report that if there are a limited number of ways to say something, you are permitted to express yourself in the same way as others without infringing (known as the "merger doctrine").
The Trouble With Freedom
The trouble with all these "freedoms" -- the merger doctrine, the right to use facts, and even fair use -- is that there are no clear lines. Since the AP is not afraid to file lawsuits you may want to proceed with caution especially if your work may be perceived as competing. If you're particularly paranoid about lawsuits, just take the facts. By the way, speaking of copyright justice, William Patry (the hardest working man in copyright) has a new book and new blog.
