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