  We derive the $T\overline{T}$-perturbed version of two-dimensional
$q$-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the
unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity.
We show that the $T\overline{T}$-deformation results in a breakdown of the
connection with a Chern-Simons theory on a Seifert manifold, and of the large
$N$ factorization into chiral and anti-chiral sectors. For the $U(N)$ gauge
theory on the sphere, we show that the large $N$ phase transition persists, and
that it is of third order and induced by instantons. The effect of the
$T\overline{T}$-deformation is to decrease the critical value of the 't Hooft
coupling, and also to extend the class of line bundles for which the phase
transition occurs. The same results are shown to hold for $(q,t)$-deformed
Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the
large $N$ limit of Yang-Mills theory, showing that the
$T\overline{T}$-deformation decreases the contribution of the Boltzmann
entropy.
