  We study the one parameter family of Fredholm determinants $\det(I-\gamma
K_{\textnormal{csin}}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator
$K_{\textnormal{csin}}$ acting on the interval $(-s,s)$ whose kernel is a cubic
generalization of the sine kernel which appears in random matrix theory. This
Fredholm determinant appears in the description of the Fermi distribution of
semiclassical non-equilibrium Fermi states in condensed matter physics as well
as in random matrix theory. Using the Riemann-Hilbert method, we calculate the
large $s$-asymptotics of $\det(I-\gamma K_{\textnormal{csin}})$ for all values
of the real parameter $\gamma$.
