In~\cite{ Johnson:2019mdp}, Clifford put forward that the specific heat at constant volume $C_{ V}$ is always negetive for the super-entropy black hole. Later in~\cite{ Cong:2019bud}, Robert et al. found $C_{ V}>0$ in certain region for the generalized exotic BTZ black holes. Futhermore, they proposed a new conjecture that as the $C_{ V}>0$, the specific heat at constant pressure $C_{ P}$ always satisfies $C_{ P}<0$ in super-entropy black hole. In this paper, we examine the both conjectures with regard to the thermodynamic instability of super-entropy black hole. By means of a super-entropy black hole, the Kerr-Newman-AdS black hole in a coordinate system that rotates at infinity, we detailedly analyze its character and notice there exists the region where $C_{ V}>0$ and meanwhile $C_{ P}>0$. Hence, we find a counterexample to the two conjectures.
Categories: gr-qc