  The aim of this paper is to study the local and the global thermodynamic
properties of the 3-dimensional rotating Gauss-Bonnet black hole. To this end
we consider the conditions for local and global thermodynamic stability of the
solution in a given ensemble of state quantities. Concerning the local analysis
we found the regions of stability for every physical specific heat together
with the existing Davies curves. Another central result is the generalization
of the notion of global thermodynamic stability, known from the standard
thermodynamics, to describe the global equilibrium of black holes. The new
approach consists of applying specific Legendre transformation of the energy or
the entropy to find the natural thermodynamic potential for the given ensemble
of macro parameters. The global stability analysis, restricted to the week
positivity conjecture is based on the properties of the new thermodynamic
potential. The advantage of this method is that it allows one to chose
different potentials, corresponding to different constraints to which the
system may be subjected. Finally, we find it natural to impose global
thermodynamic stability only where local one exists for the given black hole
solution.
