What I've done when I see stuff, is document. I don't always do anything, but I take screen shots and save in a folder. I would be upset if I saw that search and knew that he had searched her numerous times before. And I would probably confront him about it.
I agree with you on something happening again - I've told H the same thing, once more and I'm done for good.
M 16 yrs, WH62, P54 3 adult blended kids EA 11/13, BD1 6/14 PA fall 14, BD2 2/15 Piecing 2015, BD3 12/15 Separated 4/16 WH moved OW in 5/16 Divorced 6/15/17