  In this report, a general method to extract thermodynamic quantities from
solutions of the Einstein equation is developed. In 1994, Wald established that
the entropy of a black hole could be identified as a Noether charge associated
with a Killing vector of a global space-time (pseudo-Riemann) manifold. We
reconstruct Wald's method using geometrical language, e.g$.$, via differential
forms defined on the local space-time (Minkowski) manifold. Concurrently, the
abstract thermodynamics are also reconstructed using geometrical terminology,
which is parallel to general relativity. The correspondence between the
thermodynamics and general relativity can be seen clearly by comparing the two
expressions. This comparison requires a modification of Wald's method.
  The new method is applied to Schwarzschild, Kerr, and Kerr--Newman black
holes and de Sitter space. The results are consistent with previous results
obtained using various independent methods. This strongly supports the validity
of the area theorem for black holes.
