  Recently, Hawking radiation has been treated, by Robinson and Wilczek, as a
compensating flux of the energy momentum tensor required to cancel gavitational
anomaly at the event horizon(EH) of a Schwarzschild-type black hole. In this
paper, motivated by their work, Hawking radiation from the event horizon(EH)
and the de Sitter cosmological horizon(CH) of black holes in de Sitter spaces,
specifically including the purely de Sitter black hole, and the static,
spherically symmetric Schwarzschild-de Sitter black hole as well as the
rotating Kerr-de Sitter black hole, has been studied by anomalies. The result
shows that the gauge current and energy momentum tensor fluxes, required to
restore gauge invariance and general coordinate covariance at the EH and the
CH, are precisely equal to those of Hawking radiation from the EH and the CH,
respectively. It should be noted that, gauge and gravitational anomalies taken
place at the cosmological horizon(CH) arise from the fact that the effective
field theory is formulated inside the CH to integrate out the classically
irrelevant outgoing modes at the CH, which is different from those at the black
hole horizon(EH).
