I know there is some controversy about this song, but its still a classic.

Cold Beer calling my Name

This is the first song that sprang to mind - "What's cooler than being cool? Ice cold" 😃

And this was second:

A good old cowboy song!

Cold Cold Heart https://www.youtube.com/watch?v=QumMB56s4Us

And the Beach Boys did one too - only theirs has an extra 'cool' in the title...

3 Cool Cats https://www.youtube.com/watch?v=Dp3utdaB9yM

Hey Jude https://www.youtube.com/watch?v=mQER0A0ej0M

For well you know that it's a fool who plays it cool
By making his world a little colder.

Beer Never Broke My Heart https://www.youtube.com/watch?v=eKrJ6hb5Poo

Long neck ice cold beer never broke my heart...

We used to line dance to this one...

First to mind. Cold is in the lyrics.

Theme song for Cool Hand Luke: Main Title

https://www.youtube.com/watch?v=f2cKAbqWMLg

"... 'Cause if you get to close, you know I'm gone like a cool breeze."

Thought of more...

Stay cool, Boy...

In the cool of the evening...

Iceberg = cold - warning, this one from my parenting days makes me cry every time

This was fun -- chilly down, wild child! demo and movie versions

One of my Mom's favorite song....

Beneath it's snowy mantle cold and clean....

I don't remember this song, but we saw Jethro Tull in concert....a LONG time ago!

Maybe not everyone's cup of tea, but my favorite show....Riverdance.

How can the small flowers grow
If the wild winds blow
And the cold snow is all around?

In the ”Baby, It’s Cold Outside,” I think I recognized the men, Ricardo Montalban and Red Skelton, but who were the women?

Esther Williams and Betty Garrett

For some reason I can't think of songs for this theme, but I googled 'Alaska cold' since I live in Alaska:) and found this funny one

Too Cool for School https://www.youtube.com/watch?v=VzbEtG6wElA