Comments on the negative specific heat of the $T \bar T$ deformed symmetric product CFT

We show that the $T \bar T$ deformation of conformal field theories whose entropy grows as $S(E) \sim E^\gamma$ for $\gamma > 1/2$ exhibits negative specific heat in its microcanonical thermodynamic function $S({\cal E})$. We analyze the large $N$ symmetric product CFT as a concrete example of a CFT with this property and compute the thermodynamic functions such as $S({\cal E})$ and ${\cal E}(T)$. The negative specific heat in the microcanonical data is interpreted as signaling the first order phase transition when the system is coupled to a heat bath.