TT¯ $$ T\overline{T} $$ deformed partition functions

Abstract We demonstrate the presence of modular properties in partition functions of TT¯ $$ T\overline{T} $$ deformed conformal field theories. These properties are verified explicitly for the deformed free boson. The modular features facilitate a derivation of the asymptotic density of states in these theories, which turns out to interpolate between Cardy and Hagedorn behaviours. We also point out a sub-sector of the spectrum that remains undeformed under the TT¯ $$ T\overline{T} $$ flow. Finally, we comment on the deformation of the CFT vacuum character and its implications for the holographic dual