Nitpick: yes, it can.
Deep water can’t get colder than 4°C before all the water above the deep water cools down to 4°C, but then it can get colder, and even freeze, no matter how deep, if it keeps losing heat. A planet-sized ball of ice is conceivable.
So what’s the standard name for \beta? Or is it just called “beta”?
Non-transitive dice are a fun thing. Though we all have a more familiar example in the form of Rock-Paper-Scissors.
“Inverse temperature” is most common. Once introduced in a document, of course, just “\beta” suffices.
I’ve seen “coldness” used, but I probably wouldn’t use it.
Yeah, it’s a cute name that has been proffered here and there, but no one actually uses it.