  In the light of recent interest in quantum gravity in de Sitter space, we
investigate semi-classical aspects of 4-dimensional Schwarzschild-de Sitter
space-time using the method of complex paths. The standard semi-classical
techniques (such as Bogoliubov coefficients and Euclidean field theory) have
been useful to study quantum effects in space-times with single horizons;
however, none of these approaches seem to work for Schwarzschild-de Sitter or,
in general, for space-times with multiple horizons. We extend the method of
complex paths to space-times with multiple horizons and obtain the spectrum of
particles produced in these space-times. We show that the temperature of
radiation in these space-times is proportional to the effective surface gravity
-- inverse harmonic sum of surface gravity of each horizon. For the
Schwarzschild-de Sitter, we apply the method of complex paths to three
different coordinate systems -- spherically symmetric, Painleve and Lemaitre.
We show that the equilibrium temperature in Schwarzschild-de Sitter is the
harmonic mean of cosmological and event horizon temperatures. We obtain
Bogoliubov coefficients for space-times with multiple horizons by analyzing the
mode functions of the quantum fields near the horizons. We propose a new
definition of entropy for space-times with multiple horizons analogous to the
entropic definition for space-times with a single horizon. We define entropy
for these space-times to be inversely proportional to the square of the
effective surface gravity. We show that this definition of entropy for
Schwarzschild-de Sitter satisfies the D-bound conjecture.
