  Motivated by possible relation between Born-Infeld type nonlinear
electrodynamics and an effective low-energy action of open string theory,
asymptotically Reissner--Nordstrom black holes whose electric field is
described by a nonlinear electrodynamics (NLED) are studied. We take into
account a four dimensional topological static black hole ansatz and solve the
field equations, exactly, in terms of the NLED as a matter field. The main goal
of this paper is investigation of thermodynamic properties of the obtained
black holes. Moreover, we calculate the heat capacity and find that the
nonlinearity affects the minimum size of stable black holes. We also use
Legendre-invariant metric proposed by Quevedo to obtain scalar curvature
divergences. We find that the singularities of the Ricci scalar in
Geometrothermodynamics (GTD) method take place at the Davies points.
