I don't know zilch about the effectiveness of the process. What I do understand is the reasoning behind the idea of cryogenic treatment: Supposedly, it's a stress-relieving action. Stress-relieving is alignment within the crystalline structure of a metal so that changes in temperature do not lead to internal pressure changes (stress). Pardon my weird little mind, but I guess you could think of it as changing a piece of gnarled wood into a straight-grained plank.
So, if this works, it would mean that the rise in temperature of a barrel during a string of shots would not change anything in the straight line of the axis of the bore.
Like I say, I dunno. Historically, heating to some appropriate temperature and then controlling the rate of cooling to ambient temperature has been the primary method of relieving internal stress in any steel structure.
Personally, I'd tend to go along with kaferhaus' comments, particularly since I vaguely recall Gale McMillan's having made similar comments.
Art