Einstein-Hilbert (EH) action can be separated into a bulk and a surface term, with a specific ("holographic") relationship between the two, so that either can be used to extract information about the other. The surface term can also be interpreted as the entropy of the horizon in a wide class of spacetimes. Since EH action is likely to just the first term in the derivative expansion of an effective theory, it is interesting to ask whether these features continue to hold for more general gravitational actions. We provide a comprehensive analysis of Lagrangians of the form -gL=-gQabcdRabcd, in which Qabcd is a tensor with the symmetries of the curvature tensor, made from metric and curvature tensor and satisfies the condition cQabcd=0, and show that they share these features. The Lanczos-Lovelock Lagrangians are a subset of these in which Qabcd is a homogeneous function of the curvature tensor. They are all holographic, in a specific sense of the term, and-in all these cases-the surface term can be interpreted as the horizon entropy. The thermodynamics route to gravity, in which the field equations are interpreted as TdS=dE+pdV, seems to have a greater degree of validity than the field equations of Einstein gravity itself. The results suggest that the holographic feature of EH action could also serve as a new symmetry principle in constraining the semiclassical corrections to Einstein gravity. The implications are discussed. © 2006 The American Physical Society.