I'm surprised most are trying to bring thermodynamics and math into this, as the question you posed is actually a linguistics problem (mainly to do with the relation between language and culture).

The word 'twice' actually comes from the Middle English word twies, which comes from the Old Saxon word tuuīo, [1] from which the Modern English expression 'to each their own' is derived, a bastardization of its German equivalent. [2] So you actually have to look at the development of British English through the eyes of a historian knowledgeable in the Anglo-Saxon settlement of Britain between 500-700 A.D. and look at the influences of the Germanic tribes on the development of our concepts behind phrases such as twice and it wasn't me, officer, I swear.

What your weatherman is doing when he says 'twice as cold' is actually described by the Theory of Semantic Representation developed by Heinz Kloss in 1853 [3], who was able to develop the theory because he was born in Saxony [4]. He and his team of 4 graduate students performed several triple-deaf experiments funded by the NSF to look specifically at the influence of our ancestor's past deeds on the semantic stresses inflected on our sentences today. There is much controversy regarding the theory [5] so I won't say much here, but he basically solved every problem in the entire field of Comparative Socio-linguistic Stress Determination in about a decade, so his research is actually pretty important and you should read more about it sometime.

The basic argument is that what the weatherman's saying by telling you that it will be 'twice as cold tomorrow' is that relative to his shared history with you, and taking into account that his lineage probably contains at least one Germanic male somewhere [6], he is imparting on you the assumption that you are aware that how cold the air is can be directly related to your awareness of his ancestry [7]. Sorry if that wasn't clear, but linguistics can be pretty heavy stuff, and the only reason I'm able to understand most of the concepts is because I have a PhD in it. It is also worth noting that some weathermen choose not to partake in this crude linguistic self-promotion practice, so they distance themselves from these exact linguistic tools that would cloud their delivery of weather related news (which is also generally why some professional weathermen have cooler names than others, since they get to pick their names when distancing themselves from their ancestry).

I assume you don't have German ancestry, which is why you didn't pick up on it. That's okay, though, since you now don't have to feel guilty for your ancestors ending the Roman Emperor's ability to reproduce in 410 A.D., [8] thus ending the male line, which we all know was the reason the empire collapsed.