Heat Capacity and the Violation of Scaling Laws in Gravitational System

In this paper, we examine the scaling laws in gravitational system from the perspective of free energy landscape and the scaling hypothesis. It has been found that for some special black holes, their critical exponents $(0,1,2,3)$ are beyond the mean field theory, and more surprisingly violate the scaling laws. We find that the main reason for the violation of the scaling laws is that the heat capacity at constant volume $C_V$ is 0, so the critical exponent $\alpha$ is often treated as 0, which can not be derived from the scaling hypothesis. We also find that there is a symmetry violation for the two coexistence states ${\omega }_l$ and ${\omega }_s$.