How do I authorize my current computer for Apple Books after de-authorizing all previous computers?

If you already used Deauthorize All but it still shows five authorized computers, I wouldn’t keep deauthorizing or changing your Apple Account settings.

On the Mac, open Books > Account > Authorizations > Authorize This Computer and sign in with the same Apple Account used to buy the book. Apple says that after using Deauthorize All, each computer you still use has to be authorized again.

If Books still says you’ve reached the five-computer limit after that, the authorization count may not have updated correctly on Apple’s side. In that case, I’d contact Apple Support rather than make further account changes.