Warmth and Cold are like Ying and Yang in so many of our senses, not just aural. There are warm and cold colors, warm and cold tastes, warm and cold odours, even emotions, etc. Some are loosely associated with different vibrational frequencies: colour being the easiest to understand in that simplistic sense.
Art, and the state of being a human being that art represents, needs light and shade, warmth and cold. Too much of one and we yearn for the other.
Interesting, with color - the lower frequencies (red, orange, yellow) are warm and they get colder (green, blue, indigo, violet) as the frequencies increase. Some people publish charts of what colors map to musical notes - I find them too subjective, synesthesia is a complex subject but I don't think it is universally shared. But I think a large reason why digital sound is considered cold is because it can represent much higher frequencies than typical classical analog recordings seemed to feature.
Having said that, I am very aware of Nyquist and the fact that analog tape can record frequencies well beyond digital, certainly CD. But for some reason, multiple generations of tape and whatever processes were used in classical recordings had a tendency to roll off the highs and music had a warmth that can be missing in some digital recordings.
It's just not that simple though. There are elements of harmonic distortion, and randomness from moving machinery and noise and many complex things that give us a sense of warmth. Although a 4 pole Moog filter is fairly warm, and that's fairly easy to understand ...
Opinions will probably vary - I don't think their is an ultimate definition yet.