first of all there's more than one way to skin a cat, but in the end you got a hide. It can be done with heat and of course the freezing processes. Heat works best for machining with carbon based steels. Freezing works best for non-hardening things like aluminum. (actually what the process was developed for during WWII).
Freezing was developed as a method to remove retained martensite left in steels after heat treat quench. This really is only effective with non-martensetic steels contrary to popular belief. The process is very prolonged, and may take as long as three months. (I've actually seen steel in freezers for six long months). Once you quench the piece of steel, and simply let it stand the quench process will go one for days if not even weeks (the granular structure of the metal will keep changing). Now some folks have come to the conclusion that freezing a barrel (most any steel) for two or three hours, and then bring it back up to room temperature relieves stresses. To a certain extent they are right, and to an even larger extent they also have a bridge to sell you! Perhaps to a certain extent with a 4350 heat treated barrel, but forget a 416 S.S. piece of steel. (actually most any 400 series S.S. steel). And even then you'll have to repeat the process a dozen times to see any real benifit. It's the nature of the beast! Often during a very precision machining process you will induce stress into the metal (I know some folks say the opposite), and you may have a piece normalized after rough machining. Then setup and recut the part to size (rarely removing much more than .025" all the way around). With large pieces of aluminum, magnesium, and titanium you will often put it in a freezer for a week or two, and this simply keeps the part from moving all over the place from relieved stress (that's where that idea came from). It does not machine even slightly better because you never changed the alloy. Most barrels are made of 416 stainless steel these days for a reason. It machines well. But 416 is also known as a martensetic steel; which is not a good thing. Martensite is a form of granular structure that is known to be weak in structure, but with a martensetic form of steel you can never get rid of the martensite (this is not the "retained martensite"). Once again it's the nature of the beast. What it will do though is actually shrink the steel a few tenths all the way around (maybe three tenths everywhere). The expanding and contraction of the metal will also release impurities embedded in the parent metal (sulphur for one), and a trip to the meatalurgical lab will easilly show this. If you send a used barrel, it will be returned surgically clean. During the machining process the deep hole drilling will induce a lot of stress in the metal (mostly compressive), and a fluting process will create much more stress if it's done with an end mill. You can free up some of the compressive stress via heat, and a smaller amount via a deep freeze (several times). When you cut the rifeling you also relieve some of the compressive stresses from the deep hole drill and ream processes. How much is debatable, and more related to the depth and area of the cuts. Personally I would not heat the barrel once it's gone thru heat treat, and maybe try a long deep soak in a freezer to see if it even took any out at all (won't be much). After all your not dealing with a piece of Ketos or O-6, and you can't fix anything that was cast into the initial ingot.
gary