The answer to my question is as follows. The Hamiltonian given by

$$H^{can} = \lim_{\rho \to \pi/2} (cos\rho)^{2-d}\int d^{d-1}\Omega \dfrac{h_{tt}}{16\pi G_N}$$

is a surface term, which is characteristic of gravity. In general the Hamiltonian can have a volume term which has contribution from the Hamiltonian density and a surface term which has contribution from the surface integral. However if one does an ADM decomposition of the Einstein-Hilbert action, then one finds that the Hamiltonian density is a constraint which is set to zero. Therefore there is no volume contribution to the Hamiltonian.

So one is left with the surface term only. It was shown by [Regge and Teitelboim][1] that in order to properly implement the variational principle for the Hamiltonian so as to get the equations of motion for pure gravity, the surface term of the above form is necessary. This was shown not only for asymptotically $AdS$ spacetime for which we have the above Hamiltonian but also for any spacetime with a boundary in the linked paper.

[1]: https://www.sciencedirect.com/science/article/pii/0003491674904047