  In the present paper, we study the asymptotics of the Fredholm determinant
$D(x,s)$ of the finite-temperature deformation of the sine kernel, which
represents the probability that there is no particles on the interval
$(-x/\pi,x/\pi)$ in the bulk scaling limit of the finite-temperature fermion
system. The variable $s$ in $D(x,s)$ is related to the temperature. The
determinant also corresponds to the finite-temperature correlation function of
one dimensional Bose gas. We derive the asymptotics of $D(x,s)$ in several
different regimes in the $(x,s)$-plane. A third-order phase transition is
observed in the asymptotic expansions as both $x$ and $s$ tend to positive
infinity at certain related speed. The phase transition is then shown to be
described by an integral involving the Hastings-McLeod solution of the second
Painlev\'e equation.
