We prove a phase transition for the law of large numbers and fluctuations of $\mathsf F_N$, the maximum of the free energy of the log-gamma directed polymer with parameter $θ$, maximized over all possible starting and ending points in an $N\times N$ square. In particular, we find an explicit critical value $θ_c=2Ψ^{-1}(0)>0$ ($Ψ$ is the digamma function) such that: 1. For $θ<θ_c$, $\mathsf F_N+2Ψ(θ/2)N$ has order $N^{1/3}$ GUE Tracy-Widom fluctuations. 2. For $θ=θ_c$, $\mathsf F_N= Θ(N^{1/3}(\log N)^{2/3})$. 3. For $θ>θ_c$, $\mathsf F_N=Θ(\log N)$. Using a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Virág (Commun. Math. Phys. 370, 2019), we deduce a similar phase transition for the asymptotic behavior of the smallest positive eigenvalue of the aforementioned random operator.