  Sepp\"al\"ainen and Valk\'o showed in \cite{SV} that for a suitable choice of
parameters, the variance growth of the free energy of the stationary
O'Connell-Yor polymer is governed by the exponent $2/3$, characteristic of
models in the KPZ universality class.
  We develop exact formulas based on Gaussian integration by parts to relate
the cumulants of the free energy, $\log Z_{n,t}^\theta$, to expectations of
products of quenched cumulants of the time of the first jump from the boundary
into the system, $s_0$. We then use these formulas to obtain estimates for the
$k$-th central moment of $\log Z_{n,t}^\theta$ as well as the $k$-th annealed
moment of $s_0$ for $k> 2$, with nearly optimal exponents $(1/3)k+\epsilon$ and
$(2/3)k+\epsilon$, respectively.
