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We consider the $\nabla \phi$ interface model with a uniformly convexinteraction potential possessing H\"older continuous second derivatives.Combining ideas of Naddaf and Spencer with methods from quantitativehomogenization, we show that the surface tension (or free energy) associated tothe model is at least $C^{2,\beta}$ for some $\beta>0$. We also prove afluctuation-dissipation relation by identifying its Hessian with the covariancematrix characterizing the scaling limit of the model. Finally, we obtain aquantitative rate of convergence for the Hessian of the finite-volume surfacetension to that of its infinite-volume limit.

$C^2$ regularity of the surface tension for the $\nabla \phi$ interface model