A star hour was the number of hours logged in space over a period of time.
In the 2270s, Will Decker reminded James T. Kirk that he hadn't "logged a single star hour in two and a half years." That, along with Kirk's unfamiliarity with the significant refits to the USS Enterprise which had been accomplished in the intervening time, motivated Decker to question whether their mission might be compromised were Kirk to assume command of the ship for this mission. (Star Trek: The Motion Picture)