  We study the effective string description of space-like Polyakov loop
correlators at finite temperature with the goal of describing the behaviour of
the space--like string tension in the vicinity of the deconfinement transition.
To this end we construct the partition function of the Nambu-Goto effective
string theory in presence of a compact transverse direction of length $L$ equal
to the inverse temperature. We then show that, under particular conditions, our
result can be interpreted as the partition function of the $T\bar T$
deformation of the 2d quantum field theory describing a compactified bosonic
field and that this mapping allows a deeper insight on the behaviour of the
space-like observables of the theory. In particular we show, by imposing that
the spectrum of the model obeys the inviscid Burgers equation, that the $T\bar
T$ deformations follow well defined trajectories in the parameter space
$(\sigma,L)$ of the model, where $\sigma$ is the string tension, which are
characterized by a constant value of the dimensionless compactification radius
$\rho=L\sqrt{\sigma/2\pi}$. We discuss the potential usefulness of these
results for studying the space--like string tension of the underlying Lattice
Gauge Theory and its behaviour across the deconfinement transition.
