Relation between bulk Hamiltonian in AdS and stress energy of CFT

Consider the following two situations:

1. One can define a stress energy for AdS which matches with the expectation value for the CFT stress tensor.

2. Consider bulk metric perturbations of the form:
   $$g_{munu} = g^{AdS}_{mu nu} + h_{mu nu}$$
   The boundary value of $h_{munu}$ sources the CFT stress tensor. However the gravity Hamiltonian can be written in the following form (see this):
   $$H = lim_{rho to pi/2} (cosrho)^{2-d}int d^{d-1}Omega dfrac{h_{00}}{16pi G_N}$$
   where $rho to pi/2$ denotes the boundary.

Here are my questions:

1. Is the gravity Hamiltonian the $00$th component of the AdS stress energy as defined in the paper above?
2. If that is so, isn’t it contradictory that $00$th component of the AdS stress energy sources the CFT stress energy, resulting in the matching of the AdS stress energy with the expectation value of the CFT stress energy?