Classical holographic relations and alternative boundary conditions for Lovelock gravity

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