  We provide two novel ways to compute the surface gravity ($\kappa$) and the
Hawking temperature $(T_{H})$ of a stationary black hole: in the first method
$T_{H}$ is given as the three-volume integral of the Gauss-Bonnet invariant (or
the Kretschmann scalar for Ricci-flat metrics) in the total region outside the
event horizon; in the second method it is given as the surface integral of the
Riemann tensor contracted with the covariant derivative of a Killing vector on
the event horizon. To arrive at these new formulas for the black hole
temperature (and the related surface gravity), we first construct a new
differential geometric identity using the Bianchi identity and an antisymmetric
rank-$2$ tensor, valid for spacetimes with at least one Killing vector field.
The Gauss-Bonnet tensor and the Gauss-Bonnet scalar play a particular role in
this geometric identity. We calculate the surface gravity and the Hawking
temperature of the Kerr and the extremal Reissner-Nordstr\"om holes as
examples.
