  The study of thermodynamics in the view of the Hamiltonian approach is a
newest tool to analyze the thermodynamic properties of the black holes. In this
letter, we investigate the thermodynamics of $d$-dimensional ($d>3$)
asymptotically AntideSitter black holes. A thermodynamic representation based
on symplectic geometry is introduced in this letter. We extend the
thermodynamics of $d-$dimensional charged AntideSitter black holes in the views
of a Hamiltonian approach. Firstly, we study the thermodynamics in reduced
phase space and correlate with the Schwarzschild solution. Then we enhance it
in the extended phase space. In an extended phase space the thermodynamic
equations of state are stated as constraints. We apply the canonical
transformation to analyze the thermodynamics of said type of black holes. We
plot $P$-$v$ diagrams for different dimensions $d$ taking the temperatures
$T<T_c, T=T_c $ and $T>T_c$ and analyze the natures of the graphs and the
dependencies on $d$. In theses diagrams, we point out the regions of
coexistence. We also examine the phase transition by applying "Maxwell's equal
area law" of the said black holes. Here we find the regions of coexistence of
two phases which are also depicted graphically. Finally, we derive the
"Clapeyron equation" and investigate the latent heat of isothermal phase
transition.

General Relativity and Quantum Cosmology