  Complete set of modes and the Hadamard function are constructed for a scalar
field inside and outside a sphere in (D+1)-dimensional de Sitter spacetime
foliated by negative constant curvature spaces. We assume that the field obeys
Robin boundary condition on the sphere. The contributions in the Hadamard
function induced by the sphere are explicitly separated and the vacuum
expectation values (VEVs) of the field squared and energy-momentum tensor are
investigated for the hyperbolic vacuum. In the flat spacetime limit the latter
is reduced to the conformal vacuum in the Milne universe and is different from
the maximally symmetric Bunch-Davies vacuum state. The vacuum energy-momentum
tensor has a nonzero off-diagonal component that describes the energy flux in
the radial direction. The latter is a purely sphere-induced effect and is
absent in the boundary-free geometry. Depending on the constant in Robin
boundary condition and also on the radial coordinate, the energy flux can be
directed either from the sphere or towards the sphere. At early stages of the
cosmological expansion the effects of the spacetime curvature on the
sphere-induced VEVs are weak and the leading terms in the corresponding
expansions coincide with those for a sphere in the Milne universe. The
influence of the gravitational field is essential at late stages of the
expansion. Depending on the field mass and the curvature coupling parameter,
the decay of the sphere-induced VEVs, as functions of the time coordinate, is
monotonic or damping oscillatory. At large distances from the sphere the
fall-off of the sphere-induced VEVs, as functions of the geodesic distance, is
exponential for both massless and massive fields.
