  We investigate the high-temperature behavior of the directed polymer model in
dimension $1+2$. More precisely we study the difference $\Delta
\mathtt{F}(\beta)$ between the quenched and annealed free energies for small
values of the inverse temperature $\beta$. This quantity is associated to
localization properties of the polymer trajectories, and is related to the
overlap fraction of two replicas. Adapting recent techniques developed by the
authors in the context of the disordered pinning model (Berger and Lacoin,
arXiv:1503.07315 [math-ph]), we identify the sharp asymptotic high temperature
behavior \[\lim_{\beta\to 0} \, \beta^2 \log \Delta \mathtt{F}(\beta) = -\pi \,
.\]
