  We study the thermodynamical properties of electrically charged black hole
solutions of a nonlinear electrodynamics theory defined by a power p of the
Maxwell invariant, which is coupled to Einstein gravity in four and higher
spacetime dimensions. Depending on the range of the parameter p, these
solutions present different asymptotic behaviors. We compute the Euclidean
action with the appropriate boundary term in the grand canonical ensemble. The
thermodynamical quantities are identified and in particular, the mass and the
charge are shown to be finite for all classes of solutions. Interestingly, a
generalized Smarr formula is derived and it is shown that this latter encodes
perfectly the different asymptotic behaviors of the black hole solutions. The
local stability is analyzed by computing the heat capacity and the electrical
permittivity and we find that a set of small black holes are locally stable. In
contrast to the standard Reissner-Nordstrom solution, there is a first-order
phase transition between a class of these non-linear charged black holes and
the Minkowski spacetime.
