Recently, conjectures have been made that there is a correlation between thermodynamic instability and super-entropy~\cite{ { Johnson:2019mdp},{ Cong:2019bud}}. In this paper, we started with torus-like black holes and verified the correlation mentioned above. After solving the specific analytical solutions of the specific heat capacity at constant volume $C_{ V}$, the specific heat capacity at constant pressure $C_{ P}$ and plotting the figures, we obtained the results which are consistent with the hypothesis from Ref. \cite{ Cong:2019bud}. To further verify the generality of conjectures, we explored the BTZ black hole in nonlinear electrodynamics and phantom AdS black hole. And then the results are summarized via a table. Ultimately, the comparison among these three black holes leads us to the conclusion that there is no essential connection between thermodynamic instability and super-entropy.
Categories: gr-qc