  We study the Cauchy directed polymer model on $\mathbb{Z}^{1+1}$, where the
underlying random walk is in the domain of attraction to the $1$-stable law. We
show that, if the random walk satisfies certain regularity assumptions and its
symmetrized version is recurrent, then the free energy is strictly negative at
any inverse temperature $\beta>0$. Moreover, under additional regularity
assumptions on the random walk, we can identify the sharp asymptotics of the
free energy in the high temperature limit, namely, \begin{equation*}
\lim\limits_{\beta\to0}\beta^{2}\log(-p(\beta))=-c. \end{equation*}
