We investigate the thermodynamic geometries of the most general static, spherically symmetric, topological black holes of the Ho\v{r}ava--Lifshitz gravity. In particular, we show that a Legendre invariant metric derived in the context of geometrothermodynamics for the equilibrium manifold reproduces correctly the phase transition structure of these black holes. Moreover, the limiting cases in which the mass, the entropy or the Hawking temperature vanish are also accompanied by curvature singularities which indicate the limit of applicability of the thermodynamics and the geometrothermodynamics of black holes. The Einstein limit and the case of a black hole with flat horizon are also investigated.

We investigate the dependence of thermodynamic properties of black holes on the choice of statistical ensemble for a particular class of Einstein-Maxwell-Gauss-Bonnet black holes with cosmological constant. We use partial Legendre transformations in the thermodynamic limit in order to compare the results in different ensembles, and show that the phase transition structure depend on the choice of thermodynamic potential. This result implies that thermodynamic metrics which are partially Legendre invariant cannot be employed to describe black hole thermodynamics, and partly explains why a particular thermodynamic metric has been used so far in the framework of black hole geometrothermodynamics.

We present a systematic and consistent construction of geometrothermodynamics by using Riemannian contact geometry for the phase manifold and harmonic maps for the equilibrium manifold. We present several metrics for the phase manifold that are invariant with respect to Legendre transformations and induce thermodynamic metrics on the equilibrium manifold. We review all the known examples in which the curvature of the thermodynamic metrics can be used as a measure of the thermodynamic interaction.

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.