New Types of Thermodynamics from $(1+1)$-Dimensional Black Holes

For normal thermodynamic systems superadditivity $\mathrm{\S }$, homogeneity \H and concavity $\text{C}$ 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 $\text{C}$ always holds, \H is always violated and $\mathrm{\S }$ 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.