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 CVβ, the specific heat capacity at constant
pressure CPβ 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.Comment: 14 pages, 5 figures, 1 tabl