On the Thermodynamical Black Hole Stability in the Space-time of a Global Monopole in $f(R)$-Gravity

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.