  We investigate the thermodynamic properties of 5D static and spherically
symmetric black holes in (i) Einstein-Maxwell-Gauss-Bonnet theory, (ii)
Einstein-Maxwell-Gauss-Bonnet theory with negative cosmological constant, and
in (iii) Einstein-Yang-Mills-Gauss-Bonnet theory. To formulate the
thermodynamics of these black holes we use the Bekenstein-Hawking entropy
relation and, alternatively, a modified entropy formula which follows from the
first law of thermodynamics of black holes. The results of both approaches are
not equivalent. Using the formalism of geometrothermodynamics, we introduce in
the manifold of equilibrium states a Legendre invariant metric for each black
hole and for each thermodynamic approach, and show that the thermodynamic
curvature diverges at those points where the temperature vanishes and the heat
capacity diverges.
