Some good info on barrel materials and their temperature tolerances...
Carbon Steel: AISI 4140
The material used for most barrels is AISI 4140 Cr-Mo steel. This is one of the most excellent steels produced for engineering applications. This alloy steel contains by wt. %: C 0.4, Mn 1.0, Cr 1.0, and a number of other elements in lesser amounts. This steel, and its popular cousin 4340, are very popular because they have excellent properties and they are very hardenable. Hardenability is a measure of the depth of material that can be hardened during a quenching operation. High hardenability means more uniform strength and possibly faster throughput. This steel can be made fantastically strong (UTS 200 Ksi) and hard (Rockwell C 57). This material is used to make bearing balls, and if you think about it a barrel is basically a bearing, so this makes sense. This steel is easily machined, welded, otherwise handled.
One thing it also does well is oxidize. Carbon steel barrels must be protected from oxidation, since iron oxide (rust) does not form a stable protection layer on the surface like chromium. Typically barrels are "blued", which is the designation for any of the various mixtures of chemicals used to change the color of the steel to a dark blue/grey color. The blue color is actually oxidized iron, but it is Fe3O4, which is a stable layer much like the one that forms on chromium or nickel. Bluing is also done on drill bits, and that is a very high wear application. Blue, or multicolored sheen, can be seen on many heat treated parts such as hacksaw blades. This grows on the surface during heat treating. The color is dependent on the thickness, like a soap film. Bluing is not perfect, and standard Fe2O3 rust can also form on a blued firearm. Parkerizing is a phosphating process very similar to bluing. This finish is matte (black usually) and more durable than bluing, and is preferred by the military for these reasons. Also, since it does not leave a polished appearance, the parts can be beadblasted or sandblasted in preparation, increasing throughput. Incidentally, both of these processes can be done by a hobbiest with a large stainless tub full of boiling water and some chemicals which are easily purchased online.
Stainless Steel: AISI 416
Stainless steels have come a long way in this century, and 416 stainless is one of the best available. It can be heat treated to be very strong (200 ksi) and hard (>50 Rc). It is magnetic (some stainless materials are not, that is all beginning with a 3xx) and easily machined due to a high sulfur content. 416 SS contains (other than iron): <0.15% C, 12.0-14.0% Cr, <1.25% Mn, <1.0% Si, <0.06% P, >0.15% S. It is the high Cr content which makes it rust resistant, because the chromium forms a hard stable oxide which protects the metal. This layer is only a few molecules thick, so it is hardly detectable. The high machinability comes at a slight cost of oxidation resistance. The sulphur also prevents galling and binding which makes this material a good candidate for bearings.
Martensitic (400 series) SS has a marked ductile/brittle transition at cold temperatures. Actually, the jump in properties is right near room temperature, or below depending on what annealed/tempered state it is in. When the material is brittle, it is still strong, it is just more sensitive to cracking and fracture. This is most likely a problem for high impact applications rather than high strength applications. Wear is a complicated combination of strength and hardness considerations, and considering the high speed and shock of the load due to a bullet, firing a stainless rifle in the cold could potentially be a problem