  We study the roughening properties of the anharmonic elastic interface in the
presence of temporally correlated noise. The model can be seen as a
generalization of the anharmonic Larkin model, recently introduced by Purrello,
Iguain, and Kolton [Phys. Rev. E {\bf 99}, 032105 (2019)], to investigate the
effect of higher-order corrections to linear elasticity in the fate of
interfaces. We find analytical expressions for the critical exponents as a
function of the anharmonicity index $n$, the noise correlator range $\theta
\in[0,1/2]$, and dimension $d$. In $d=1$ we find that the interface becomes
faceted and exhibits anomalous scaling for $\theta > 1/4$ for any degree of
anharmonicity $n > 1$. Analytical expressions for the anomalous exponents
$\alpha_\mathrm{loc}$ and $\kappa$ are obtained and compared with a numerical
integration of the model. Our theoretical results show that anomalous
roughening cannot exist for this model in dimensions $d > 1$.
