  We investigate the vacuum energy density induced by quantum fluctuations of a
bulk scalar field with general curvature coupling parameter on two codimension
one parallel branes in a $(D+1)$-dimensional background spacetime
${\mathrm{AdS}}_{D1+1}\times \Sigma $ with a warped internal space $\Sigma $.
It is assumed that on the branes the field obeys Robin boundary conditions.
Using the generalized zeta function technique in combination with contour
integral representations, the surface energies on the branes are presented in
the form of the sums of single brane and second brane induced parts. For the
geometry of a single brane both regions, on the left (L-region) and on the
right (R-region), of the brane are considered. The surface densities for
separate L- and R-regions contain pole and finite contributions. For an
infinitely thin brane taking these regions together, in odd spatial dimensions
the pole parts cancel and the total surface energy is finite. The parts in the
surface densities generated by the presence of the second brane are finite for
all nonzero values of the interbrane separation. The contribution of the
Kaluza-Klein modes along $\Sigma $ is investigated in various limiting cases.
It is shown that for large distances between the branes the induced surface
densities give rise to an exponentially suppressed cosmological constant on the
brane. In the higher dimensional generalization of the Randall-Sundrum
braneworld model, for the interbrane distances solving the hierarchy problem,
the cosmological constant generated on the visible brane is of the right order
of magnitude with the value suggested by the cosmological observations.
