  It is well-known that black hole can be endowed with topological charge
coming as a result of phase transition(s) in the early universe. It has also
recently been revealed that such object might exist in one of the modified
theory of gravity called the Eddington-inspired-Born-Infeld (EiBI) theory. Here
we shall investigate the 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 inifinities 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 the black hole mechanics and the Smarr's law for
black holes with $\Lambda\neq0$. Related to the second feature, we obtained the
forbidden range of parameter space for EiBI AdS black hole with global
monopole, which corresponds to $\frac{-(1 - \Delta)}{\Lambda } \leq \kappa \leq
-\frac{1}{\Lambda }$. The dependencies on $\Lambda$, the EiBI constant $\kappa$
and the global monopole charge $\eta$ 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.
