  For normal thermodynamic systems superadditivity $\S$, homogeneity $\H$ and
concavity $\C$ of the entropy hold, whereas for $(3+1)$-dimensional black holes
the latter two properties are violated. We show that $(1+1)$-dimensional black
holes exhibit qualitatively new types of thermodynamic behaviour, discussed
here for the first time, in which $\C$ always holds, $\H$ is always violated
and $\S$ may or may not be violated, depending of the magnitude of the black
hole mass. Hence it is now seen that neither superadditivity nor concavity
encapsulate the meaning of the second law in all situations.
