  We investigate thermodynamic curvatures of the Kerr and Reissner-Nordstr\"om
(RN) black holes in spacetime dimensions higher than four. These black holes
possess thermodynamic geometries similar to those in four dimensional
spacetime. The thermodynamic geometries are the Ruppeiner geometry and the
conformally related Weinhold geometry. The Ruppeiner geometry for $d=5$ Kerr
black hole is curved and divergent in the extremal limit. For $d \geq 6$ Kerr
black hole there is no extremality but the Ruppeiner curvature diverges where
one suspects that the black hole becomes unstable. The Weinhold geometry of the
Kerr black hole in arbitrary dimension is a flat geometry. For RN black hole
the Ruppeiner geometry is flat in all spacetime dimensions, whereas its
Weinhold geometry is curved. In $d \geq 5$ the Kerr black hole can possess more
than one angular momentum. Finally we discuss the Ruppeiner geometry for the
Kerr black hole in $d=5$ with double angular momenta.

High Energy Physics - Theory;General Relativity and Quantum Cosmology