Never sent out gun parts for cryo, but have sent some hot rod parts out.
What’s the benefit of sending out a barrel?
I sent out things like axleshafts, transmission outputs and yokes because it would align the grain structure. They were usually CrMo or 300M material to maximize those benefits.
I went to the woods because I wished to live deliberately...
I had a 7mm BR barrel for a silhouette pistol cyro treated years ago. Started with a Shilen Select Stainless blank, turned it, chambered it and spiral fluted it. Sent it out and had it treated. One of the most accurate pistols I've ever shot but the most impressive thing is how easy it is to clean. One or two passes with a brush then a couple of patches and you're good to go. Over 3000 rounds and virtually no throat erosion. I'm a believer.
Freezing the barrel and other parts will take all the stress out of the steel and re align the atoms equally through out the part. Think about this. If you took a torch and heated one end of the barrel and started down the side, the heat will pull the molecules with the hottest part of the torch. If you put it in a oven to take the stress out of it, you are doing it with more heat. When you freeze I like -300 , it will shrink it a little close up all micro fractures, and re align the structure on a molecular level. Now there is always people saying heat is better. I have had better results from cold temps.