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Abstract: We investigate the intrinsic thermal Hall conductivity contributed by optical phonons in a cubic system. The discrete rotational symmetry of the system splits the degeneracy of transverse modes across most regions of wave-vector space, except along a few high-symmetry lines. Consequently, in the presence of an external magnetic field, phonon Berry curvatures become sharply peaked near these high-symmetry lines. We find that the singular distribution of the Berry curvature induces an intrinsic thermal Hall conductivity that is significantly enhanced compared to an isotropic system. It exhibits a nonlinear $B\ln B$ dependence on the magnetic field $B$ and a non-monotonic temperature dependence. At elevated temperatures, it reverses sign and approaches a non-vanishing value asymptotically. Our analysis indicates that the behavior results from competition between contributions from Berry curvatures near different high-symmetry lines.