  We report temperature-dependent surface x-ray scattering studies of the
orbital ordered surface in La$_{0.5}$Sr$_{1.5}$MnO$_4$. We find that the
interfacial width of the electronic order grows as the bulk ordering
temperature is approached from below, though the bulk correlation length
remains unchanged. Close to the transition, the surface is so rough that there
is no well-defined electronic surface, despite the presence of bulk electronic
order, that is the electronic surface has melted. Above the bulk transition,
finite-sized isotropic fluctuations of orbital order are observed, with a
correlation length equal to that of the electronic surfaces' in-plane
correlation length at the transition temperature.
