The Painleve-Gullstrand coordinates are extended to describe the black hole
in the cosmological environment: the Schwarzschild-de-Sitter black hole, which
has two horizons. The extension is made using the Arnowitt-Deser-Misner
formalism. In this extension, which describes the metric in the whole range of
radial coordinates $0<r < \infty$, there is the point $r=r_0$ at which the
shift function (velocity) changes sign. At this point the observer is at rest,
while the observers at $r<r_0$ are free falling to the black hole and the
observers at $r>r_0$ are free falling towards the cosmological horizon. The
existence of the stationary observer allows to determine the temperature of
Hawking radiation, which is in agreement with R. Bousso and S.W. Hawking, Phys.
Rev. D 54, 6312 (1996). It is the red-shifted modification of the conventional
Hawking temperature determined by the gravity at the horizon. We also consider
the Painlev\’e-Gullstrand coordinates and their extension for such
configurations as Schwarzschild-de-Sitter white hole, where the sign of the
shift function is everywhere positive; the black hole in the environment of the
contracting de Sitter spacetime, where the sign of the shift function is
everywhere negative; and the white hole in the contracting de Sitter spacetime,
where the shift velocity changes sign at $r=r_0$.
