The torque spec for mounting a carbon railed saddle is not normally different from a metal-railed saddle. However, if the clamp the seatpost uses to attach to the saddle rails is not compatible with ovalized carbon rails and one simply torques the seatpost to rails clamp up to spec, damage can occur.
Carbon railed seats do require one to be careful as to the type attachment employed by the seatpost. This is driven by the fact the rails are carbon and if damaged/cracked/crushed it cannot be repaired AND that the rails on most (all?) carbon saddles do not have a round cross-section but actually an oval one. And the Brooks C13 Carbon is no different.
From the Brooks website, the spec on the rails for the C13 Carbon are: Rail: 7x9 mm Carbon Braided. 7x9 is oval, whereas metal rails (titanium, steel, other alloys, etc.) are round (7mm typically).
The Brooks website is silent on seatpost compatibility with ovalized carbon rails. At least I could not find anything.
Fizik (saddles), however, has been more prudent in communicating about seatpost compatibility with their carbon railed saddles (most of theirs are 7x9 mm as well). Fizik may have seen some early rail failures due to seatpost designs that caused stresses in the rails that created the failures. The current guidance Fizik gives is "Before installation, verify compatibility with your seatpost manufacturer between saddle rails and the seatpost itself. Many seatpost manufacturers provide different clamping solutions to assure full compatibility."
Less than a year ago, they offered more detail (no longer available on Fizik website):
Q: Is my seatpost compatible with your carbon braided rail saddles?
A: Most seatposts currently on the market are compatible with our
7x9mm carbon rails. As a guide: all those fixing the saddle from the
top to the bottom. (But the ones that are not compatible, are
typically the types that fix the saddle from the sides.)
There are 5 types of seatpost clamps represented:
Type A: Single Bolt Top-to-Bottom clamp
Type B: Twin Bolt (Front/Rear) Top-to-Bottom clamp
Type C: Twin Bolt (Right/Left) Top-to-Bottom clamp
Type D: Side-to-Side clamp (single bolt)
Type E: Hinged Rotational Clamp
Type A, B, & C are typically COMPATIBLE with 7x9 mm saddle rails.
If your seatpost is compatible, it still is prudent to be sure that any seat rail clamp you use is free of sharp edges that could gouge and damage the carbon.
Finally, there was a Bike Exchange question that is related to this aspect from a few years ago: Oval carbon saddle rails compatibility
Hope this information can help prevent damage if your seatpost is not compatible.