It is well known that a black hole can be endowed with a topological charge coming as a result of phase transition(s) in the early universe. It has also recently been revealed that such an object might exist in one of the modified theory of gravity called the Eddington-inspired-Born-Infeld (EiBI) theory. Here we shall investigate the possibility of phase transitions, and in general thermodynamic phenomena, of EiBI-anti-de Sitter (AdS) black hole with (topologically-charged) global monopole. This is the first work that applies the full Euclidean action formalism in this model. We provide a counterterm to cancel infinities and argue that it is the most suitable among other possibilities. Our investigation reveals that the state variables obtained are found to obey the first law of black hole mechanics and Smarr's law for black holes with Λ≠0. Related to the second feature, we obtained the forbidden range of parameter space for EiBI AdS black hole with global monopole, which corresponds to -(1-Δ)Λ≤κ≤-1Λ. The dependencies on Λ, the EiBI constant κ and the global monopole charge η of the state variables and state functions obtained manifests in a Schwarzchild AdS-like phase transition for black holes with parameters below the lower bound of the forbidden range, and could also manifest in a Schwarzschild-flat like phase behavior for black holes with parameters above the higher bound of the forbidden range.