What books are needed to run book one of this adventure path?
Technically, none, since the Archives of Nethys (Official SRD) publish pretty much all of the Pathfinder rules online. However, it is somewhat difficult to learn the Pathfinder rules at the table purely by using the Archives as reference.
Sight unseen. I would say for sure the Core Rulebook. A Bestiary might be good but I don't know which of the 3 will have the most relevant monsters plus those are actually really easy to look up on the Archives.
The AP does reference rules from the Dark Archive (a book, not to be confused with the SRD Archives of Nethys ofc), specifically deviant abilities, however the entirety of the deviant abilities mechanics should be freely available in the player's guide for the adventure, which has already been released as a pdf you can pick up probably by searching Gatewalkers Player's Guide.
Book 1 says the CRB is the only book needed. Deviant rules is in the players guide. Bestiary can be found in a plethora of places like Demiplanes pathfinder nexus.