  We develop a quenched thermodynamic formalism for open random dynamical
systems generated by finitely branched, piecewise-monotone mappings of the
interval. The openness refers to the presence of holes in the interval, which
terminate trajectories once they enter; the holes may also be random. Our
random driving is generated by an invertible, ergodic, measure-preserving
transformation $\sigma$ on a probability space $(\Omega,\mathscr{F},m)$. For
each $\omega\in\Omega$ we associate a piecewise-monotone, surjective map
$T_\omega:I\to I$, and a hole $H_\omega\subset I$; the map $T_\omega$, the
random potential $\varphi_\omega$, and the hole $H_\omega$ generate the
corresponding open transfer operator $\mathcal{L}_\omega$. For a contracting
potential, under a condition on the open random dynamics in the spirit of
Liverani--Maume-Deschamps, we prove there exists a unique random probability
measure $\nu_\omega$ supported on the survivor set ${X}_{\omega,\infty}$
satisfying $\nu_{\sigma(\omega)}(\mathcal{L}_\omega
f)=\lambda_\omega\nu_\omega(f)$. We also prove the existence of a unique random
family of functions $q_\omega$ that satisfy $\mathcal{L}_\omega
q_\omega=\lambda_\omega q_{\sigma(\omega)}$. These yield an ergodic random
invariant measure $\mu=\nu q$ supported on the global survivor set, while $q$
combined with the random closed conformal measure yields a unique random
absolutely continuous conditional invariant measure (RACCIM) $\eta$ supported
on $I$. We prove quasi-compactness of the transfer operator cocycle and
exponential decay of correlations for $\mu$. Finally, the escape rates of the
random closed conformal measure and the RACCIM $\eta$ coincide, and are given
in terms of the expected pressure, as is the Hausdorff dimension of the
surviving set $X_{\omega,\infty}$. We provide examples of our general theory,
including random $\beta$-transformations and random Lasota-Yorke maps.
