  In this work, we re-assess a class of black hole solutions in a global
monopole spacetime in the framework of an $f(R)$-gravity model. Our main line
of investigation consists in considering a region close enough to the black
hole, but such that the weak field approximation is still valid. The stability
of the black hole is studied in terms of its thermodynamical properties, with
the radial coordinate written as a power law function with the status of the
main factor underneath the stability of the model. We obtain the explicit
expressions for the thermodynamical quantities of the black hole as functions
of the event horizon, by considering both the Hawking and the local
temperatures. The phase transitions that may occur in this system, including
the Hawking-Page phase transition, are inspected with particular attention. We
work out and contemplate a solution of special interest in which one of the
parameters is related to the cosmological constant. Our main result sets out to
establish a comparison between both the Hawking and the local formalisms for
the black hole in the framework of the $f(R)$-gravity in the particular
space-time adopted here.
