  It is well-known that the exact solution of non-linear $\sigma$ model coupled
to gravity can be perceived as an exterior gravitational field of a global
monopole. Here we study Einstein's equations coupled to a non-linear $\sigma$
model with Dirac-Born-Infeld (DBI) kinetic term in $D$ dimensions. The solution
describes a metric around a DBI global defects. When the core is smaller than
its Schwarzschild radius it can be interpreted as a black hole having DBI
scalar hair with deficit conical angle. The solutions exist for all $D$, but
they can be expressed as polynomial functions in $r$ only when $D$ is even. We
give conditions for the mass $M$ and the scalar charge $\eta$ in the extremal
case. We also investigate the thermodynamic properties of the black holes in
canonical ensemble. The monopole alter the stability differently in each
dimensions. As the charge increases the black hole radiates more, in contrast
to its counterpart with ordinary global defects where the Hawking temperature
is minimum for critical $\eta$. This behavior can also be observed for
variation of DBI coupling, $\beta$. As it gets stronger ($\beta\ll1$) the
temperature increases. By studying the heat capacity we can infer that there is
no phase transition in asymptotically-flat spacetime. The AdS black holes, on
the other hand, undergo a first-ordered phase transition in the Hawking-Page
type. The increase of the DBI coupling renders the phase transition happen for
larger radius.
