  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.
