  In this paper, we analyze the thermodynamic properties of the Anti de Sitter
black hole in the Einstein-Maxwell-Yang-Mills-AdS gravity (EMYM) via many
approaches and in different thermodynamical ensembles (canonical/ grand
canonical). First, we give a concise overview of this phase structure in the
entropy-thermal diagram for fixed charges then we investigate this
thermodynamical structure in fixed potentials ensemble. The Next relevant step
is recalling the nonlocal observables such as holographic entanglement entropy
and two-point correlation function to show that both observables exhibit a Van
der Waals-like behavior in our numerical accuracy and just near the critical
line as the case of the thermal entropy for fixed charges by checking Maxwell's
equal area law and the critical exponent. In the light of the grand canonical
ensemble, we also find a newly phase structure for such a black hole where the
critical behavior disappears in the thermal picture as well as in the
holographic one.
