  We obtain the classical holographic relation for the general Lovelock gravity
and decompose the full Lagrangian into the bulk term and the surface term,
expressed as a total derivative $\partial_\mu J^\mu$. By classical holographic
relation, we mean that $J^\mu$ is determined completely by the bulk term. We
find that the bulk term is not degenerate, or first-order in this
foliation-independent approach. We then consider the Arnowitt-Deser-Misner
(ADM) formalism where the foliation coordinate $w$ is treated as special. We
obtain the classical holographic-degenerate relation with the first-order bulk
term that does not involve higher than one derivative of $w$. For Einstein
gravity, the two approaches lead to the same bulk term, but different ones for
higher-order Lovelock gravities. The classical holographic-degenerate
formulation in the ADM approach allows us to consider alternative boundary
conditions in the variation principle with different Myers terms. We show in
the semiclassical approximation that the black hole entropy in all cases is the
same as the one obtained under the standard Dirichlet boundary condition. We
also generalize the formalism to general $f(L_{\rm Lovelock}^{(k)})$-gravity.
