We investigate the thermodynamic properties of 5D static and sphericallysymmetric black holes in (i) Einstein-Maxwell-Gauss-Bonnet theory, (ii)Einstein-Maxwell-Gauss-Bonnet theory with negative cosmological constant, andin (iii) Einstein-Yang-Mills-Gauss-Bonnet theory. To formulate thethermodynamics of these black holes we use the Bekenstein-Hawking entropyrelation and, alternatively, a modified entropy formula which follows from thefirst law of thermodynamics of black holes. The results of both approaches arenot equivalent. Using the formalism of geometrothermodynamics, we introduce inthe manifold of equilibrium states a Legendre invariant metric for each blackhole and for each thermodynamic approach, and show that the thermodynamiccurvature diverges at those points where the temperature vanishes and the heatcapacity diverges.
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