However, one can have negative temperatures (statistical mechanics version). They do not lead to negative entropies though.
Yeah, but this is a general feature of statistical models of a lot of things -- you can "borrow" locally into some strange negatives as long as it disappears in the long-term integration. Which is kind of neat in some ways.
Me, I'm stuck in the computational theoretic view. The only statistical models we like are Bayesian (zzzzz....).