"It should give him pause" meaning that should cause him to think before acting or simply give him pause for thought, is a quaint construction which sounds like C17 English. But is it new and American, old American but new here, or antiquated English which was current here in the past but which Americans never stopped using?
It seems to be a variation on "give pause to", which the Concise Oxford Dictionary defines as 'cause (person) to hesitate' [both in British and US English].
The Hamlet quote in the above link is recorded in the OED as the earliest use of the phrase. It would seem, therefore, to be just another one of the Bard's very own coinages.