  We investigate constraints imposed by entanglement on gravity in the context
of holography. First, by demanding that relative entropy is positive and using
the Ryu-Takayanagi entropy functional, we find certain constraints at a
nonlinear level for the dual gravity. Second, by considering Gauss-Bonnet
gravity, we show that for a class of small perturbations around the vacuum
state, the positivity of the two point function of the field theory stress
tensor guarantees the positivity of the relative entropy. Further, if we impose
that the entangling surface closes off smoothly in the bulk interior, we find
restrictions on the coupling constant in Gauss-Bonnet gravity. We also give an
example of an anisotropic excited state in an unstable phase with broken
conformal invariance which leads to a negative relative entropy.
