Does Temperature have a Metric Structure?

Abstract

Is there anything more to temperature than the ordering of things from colder to hotter? Are there also facts, for example, about how much hotter (twice as hot, three times as hot...) one thing is than another? There certainly are—but the only strong justification for this claim comes from statistical mechanics. What we knew about temperature before the advent of statistical mechanics (what we knew about it from thermodynamics) provided only weak reasons to believe it.