One intepretive argument we can make and easily support is that the weather represents a reality which Holden is attempting to ignore in The Catcher in the Rye.
Early in the book, Holden mentions the cold weather and introduces his red hat with the ear flaps. The purpose of the hat is to keep out the cold. Holden's attachment to the hat is irrational, as he says himself, and the attitude he holds regarding the hat is repeatedly associated with an intentional indifference.
Holden wears the hat, though it looks ridiculous. He wears the hat, though it is a sign of his irrationality.
We can point to similar behaviors in Holden in relation to the weather. He walks home from the bar to the hotel and sleeps outside despite the cold. Though he feels the cold, he does not allow it to shape his behavior.
The reason Holden chooses to ignore the weather is in keeping with his other avoidance behavior. He cannot face reality. He cannot face the fact that his brother Allie is dead. Holden persists in speaking with Allie and keeps him alive in his imagination. The reality is a cold one. Allie is dead. Yet Holden persists in behaving as if this fact is debatable or adjustable.
Holden, we might say, is essentially defenseless against the weather but tries to shut it out and clings to tokens of indifference to it, as symbolized in his red hat.