  We study the gauge invariance of physical observables in holographic theories
under the local diffeomorphism. We find that gauge invariance is intimately
related to the holographic renormalisation: the local counter terms defined in
the boundary cancel most of gauge dependences of the on-shell action as well as
the divergences. There is a mismatch in the degrees of freedom between the bulk
theory and the boundary one. We resolve this problem by noticing that there is
a residual gauge symmetry(RGS). By extending the RGS such that it satisfies
infalling boundary condition at the horizon, we can understand the problem in
the context of general holographic embedding of a global symmetry at the
boundary into the local gauge symmetry in the bulk.
