We derive an expression for conserved charges in Lovelock anti-de Sitter gravity for solutions having k-fold degenerate vacua, making manifest a link between the degeneracy of a given vacuum and the nonlinearity of the energy formula. We show for a black hole solution to the field equations on a branch of multiplicity k that its mass comes from an expression that contains the product of k Weyl tensors. We prove that all divergent contributions of the type (Weyl)q, with 1≤q<k, are suppressed. Our conserved charge definition is a natural generalization of the conformal mass by Ashtekar, Magnon and Das to the cases when k>1. Our results provide insight on the holographic properties of degenerate Lovelock theories.