  Holographic entanglement entropy and the first law of thermodynamics are
believed to decode the gravity theory in the bulk.
  In particular, assuming the Ryu-Takayanagi (RT)\cite{ryu-takayanagi} formula
holds for ball-shaped regions on the boundary around CFT vacuum states
implies\cite{Nonlinear-Faulkner} a bulk gravity theory equivalent to Einstein
gravity through second-order perturbations. In this paper, we show that the
same assumptions can also give rise to second-order Lovelock gravity.
Specifically, we generalize the procedure in \cite{Nonlinear-Faulkner} to show
that the arguments there also hold for Lovelock gravity by proving through
second-order perturbation theory, the entropy calculated using the Wald
formula\cite{Wald_noether} in Lovelock also obeys an area law (at least up to
second order). Since the equations for second-order perturbations of Lovelock
gravity are different in general from the second-order perturbation of the
Einstein-Hilbert action, our work shows that the holographic area law cannot
determine a unique bulk theory even for second-order perturbations assuming
only RT on ball-shaped regions. It is anticipated that RT on all subregions is
expected to encode the full non-linear Einstein equations on asymptotically AdS
spacetimes.
