  In this paper, we investigate the correlation functions of the conformal
field theory (CFT) with the $T\bar{T}$ deformation on torus in terms of
perturbative CFT approach, which is the extension of the previous
investigations on correlation functions defined on a plane. We systematically
obtain the first order correction to the correlation functions of the CFTs with
$T\bar{T}$ deformation in both operator formalism and path integral language.
As a consistency check, we compute the deformed partition function, namely the
zero-point correlation function, up to the first order, which is consistent
with results in literature. Moreover, we obtain a new recursion relation for
correlation functions with multiple $T$'s and $\bar{T}$'s insertion in generic
CFTs on torus. Base on the recursion relations, we study some correlation
functions of stress tensors up to the first order under $T\bar{T}$ deformation.
