  In this paper, we extend the idea that the spectrum of Hawking radiation can
reveal valuable information on a number of parameters that characterize a
particular black hole background - such as the dimensionality of spacetime and
the value of coupling constants - to gain information on another important
aspect: the curvature of spacetime. We investigate the emission of Hawking
radiation from a D-dimensional Schwarzschild-de-Sitter black hole emitted in
the form of scalar fields, and employ both analytical and numerical techniques
to calculate greybody factors and differential energy emission rates on the
brane and in the bulk. The energy emission rate of the black hole is
significantly enhanced in the high-energy regime with the number of spacelike
dimensions. On the other hand, in the low-energy part of the spectrum, it is
the cosmological constant that leaves a clear footprint, through a
characteristic, constant emission rate of ultrasoft quanta determined by the
values of black hole and cosmological horizons. Our results are applicable to
"small" black holes arising in theories with an arbitrary number and size of
extra dimensions, as well as to pure 4-dimensional primordial black holes,
embedded in a de Sitter spacetime.
