Thermodynamics of $T bar T$, $J bar T$, $T bar J$ deformed conformal field theories

We compute the Hagedorn temperature of $mu T bar T + varepsilon_+ J bar T + varepsilon_-T bar J$ deformed CFT using the universal kernel formula for the thermal partition function. We find a closed analytic expression for the free energy and the Hagedorn temperature as a function of $mu$, $varepsilon_+$, and $varepsilon_-$ for the case of a compact scalar boson by taking the large volume limit. We also compute the Hagedorn temperature for the single trace deformed $AdS_3 times S^1 times T^3 times S^3$ using holographic methods. We identify black hole configurations whose thermodynamics matches the functional dependence on $(mu, varepsilon_+, varepsilon_-)$ of the double trace deformed compact scalars.