I remember some discussion on this fairly early in the life of HBD, and people throwing out numbers showing that X starter doesn't get worse in the later innings, and in fact maybe was better, but it's really so difficult to get a meaningful sample, especially for the later innings. And further, it would take some serious work to figure out how pitchers do when they are at a given pitch count.
There has to be in game fatigue, or as has previously been alluded to, we'd have heard of instances of someone all of a sudden starting their closer in a game 7 and getting a complete game shutout of him, or something like that.