"It is not cold."
寒い is an い-adjective which means it functions like a noun. It can conjugate and end a sentence on its own since it already contains the meaning of "is X" in it. The additional です is only to make the sentence more polite. It does not conjugate.
です is only required and conjugates when it follows a noun or な-adjective which functions like a noun.
If you wanted to make it more formal/literary ない is the casual form of ありません so you can swap them