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We discuss the criteria that must be satisfied by a well-posed variationalprinciple. We clarify the role of Gibbons-Hawking-York type boundary terms inthe actions of higher derivative models of gravity, such as F(R) gravity, andargue that the correct boundary terms are the naive ones obtained though thecorrespondence with scalar-tensor theory, despite the fact that variations ofnormal derivatives of the metric must be fixed on the boundary. We show in thecase of F(R) gravity that these boundary terms reproduce the correct ADM energyin the hamiltonian formalism, and the correct entropy for black holes in thesemi-classical approximation.

Boundary Terms, Variational Principles and Higher Derivative Modified Gravity