  The Noether current and its variation relation with respect to diffeomorphism
invariance of gravitational theories have been derived from the horizontal
variation and vertical-horizontal bi-variation of the Lagrangian, respectively.
For Einstein's GR in the stationary, axisymmetric black holes, the mass formula
in vacuum can be derived from this Noether current although it definitely
vanishes. This indicates that the mass formula of black holes is a vanishing
Noether charge in this case. The first law of black hole thermodynamics can
also be derived from the variation relation of this vanishing Noether current.
