The cold never bothered me anyway.
As you know, this phrase is in the lyrics of Ana and Snow Queen.
Will you tell me why bother is changed to the past tense?
Because she is talking about past experiences. Up until now, the cold never bothered her. So why should it this time? That's the nuance of using the past tense here.
It's not too different to say "bothers" but it doesn't imply past experience and needs a reason to explain.
Imagine two friends. One is from Sweden and the other is from Jamaica. They both live in Tokyo. It's winter and they're out walking around. The Jamaican friend says, "How are you not cold? It's freezing out here!" The Swedish friend just replies, "I'm used to the cold. It never bothers me."