This geometry has been applied to black hole thermodynamics, with some physically relevant results. The most physically significant case is for the Kerr black hole in higher dimensions, where the curvature singularity signals thermodynamic instability, as found earlier by conventional methods.
The entropy of a black hole is given by the well-known Bekenstein–Hawking formula
where  is Boltzmanns constant,  the speed of light,  Newtons constant and  is the area of the event horizon of the black hole. Calculating the Ruppeiner geometry of the black hole's entropy is, in principle, straightforward, but it is important that the entropy should be written in terms of extensive parameters, 
where  is ADM mass of the black hole and  are the conserved charges and  runs from 1 to n. The signature of the metric reflects the sign of the hole's specific heat. For a Reissner-Nordström black hole, the Ruppeiner metric has a Lorentzian signature which corresponds to the negative heat capacity it possess, while for the BTZ black hole, we have a Euclidean signature. This calculation cannot be done for the Schwarzschild black hole, because its entropy is 
which renders the metric degenerate.