  In a four-dimensional Schwarzschild-de Sitter background, the spherically
symmetric ($\ell=0$) contribution to the Hadamard two-point correlation
function is computed for a massless minimally-coupled scalar field in the Unruh
state. Consideration is given to spacetime points located exterior to the black
hole horizon, but within the cosmological horizon. Previously it was found in
two dimensions for spatially separated points that the Hadamard function
exhibits unbounded linear growth at late times, with a rate of growth
proportional to the sum of the black hole and cosmological surface gravities.
Here it is shown numerically that this behavior persists in four dimensions,
where scattering effects associated with the scalar field modes lead to a
modification of the two-dimensional result. An analytic approximation is
derived for the growth rate in four dimensions and, in the limit that the black
hole vanishes, the leading order contribution is equivalent to the rate of
growth for the Hadamard function found previously for de Sitter space in
cosmological coordinates.
