As it is shown in Prove that negative absolute temperatures are actually hotter than positive absolute temperatures, a body with absolute negative temperature is hotter than a body with any absolute positive temperature, where the hotter body is the one which the heat flows from. On the other hand, the efficiency of a reversible engine is $$\eta=1-\frac{T_H}{T_C},$$ where $T_H$ and $T_C$ are the absolute temperature of the hot and cold sources, respectively. If $T_C>0$ and $T_H<0$, then $\eta>1$. But since the efficiency is the ratio between work delivered and heat absorbed from the hot source, the above result means that it is delivered more work than the heat received, $|W|>|Q_H|$, violating the first law, which says that $|W|=|Q_H|-|Q_C|$.
So, is it not possible to have a reversible engine working between negative and positive absolute temperatures? If not, what law forbids that? Or, contrary to what is shown the link, we cannot assume that heat flows from negative to positive absolute temperatures? The last alternative is that everything is fine. But then, what about the first law being violated?
