Are you sure it’s cold, and not numb/reduced sensation?
i.e. if you touch it with the good hand, or somebody else touches it, does it feel a normal temperature?
The brain tries to make the best sense it can of the information it’s given. As it does not expect your arm to be numb for no reason, and it knows cold can cause numbness, it may reason: “Aha, the arm must be cold!”
It’s very hard to counter this powerful illusion, because the brain thinks: “I must be cold” is a much more sensible explanation than: “I’ve randomly lost sensation in bits of me” - and, in healthy people, it would be right!
Some people find wearing tight clothing on the affected limb makes it feel more normal, because it gives the brain a reason sensation is altered (something tight is on it), rather than having to assume it must be cold.
The brain is both incredibly clever, yet incredibly easy to deceive, in some ways. If you can give it a perfectly good reason why it’s feeling what it’s feeling, e.g: “There’s a tight thing on my arm”, it will be much less concerned about it, because what it’s feeling makes sense in the light of known information.