  A thermodynamic study of the Kondo insulator SmB$_6$ is pursued to elucidate
the well-known anomalous low-temperature electronic-like specific heat
contribution conjectured to arise from metallic surface states. A general
thermodynamic description of topological Kondo insulators is developed using a
mean-field slave-boson approximation and an approach inspired by Hill
thermodynamics to study the phase transitions with the critical exponents of
the model. The results show consistency with the Josephson hyper-scaling
relation. We further apply this framework to SmB$_6$. By separating the bulk
and boundary contributions to the heat capacity, we show that while the surface
states contribute to an increase of the heat capacity upon lowering the
temperature, the effect is immeasurably small. This suggests that an alternate
explanation is required for the anomalous low-temperature contributions to the
specific heat in this material.
