  The Weyl tensor is the trace-free part of the Riemann tensor. Therefore, it
is independent of the energy-momentum tensor and is thus not linked to the
dynamics of gravitational fields. In this paper, we explore its possible
thermodynamical property (i.e. its relationship with the black hole entropy).
For a Schwarzschild black hole, the Weyl scalar invariant,
$C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$, is proportional to its
Bekenstein--Hawking entropy. This observation inspires us to interpret
$C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$ as the entropy density of the
gravitational fields of black holes. A dimensional analysis indicates that this
interpretation is only valid in 5-dimensional space-time. We calculate the
volume integrals of $C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$ for the
5-dimensional Schwarzschild and Schwarzschild--anti-de Sitter black holes, and
find that these integrals indeed lead to the correct entropy formulae, only up
to some coefficients.
