  We investigate whether the new horizon first law still holds in
$f(R,R^{\mu\nu}R_{\mu\nu})$ theory. For this complicated theory, we first
determine the entropy of black hole via Wald method, then we derive the energy
by using the new horizon first law, the degenerate Legendre transformation, and
the gravitational field equations. For application, we consider the
quadratic-curvature gravity and firstly calculate the entropy and the energy
for a static spherically symmetric black hole, which reduces to the results
obtained in literatures for a Schwarzschild-(A)dS black hole.
