  We propose an effective model for an exponentially expanding universe in the
brane-world scenario. The setup consists of a 5D black hole and a brane close
to the black hole horizon. In case the brane acquires a specific configuration,
which we deduce from stability arguments, the induced metric outside the black
hole horizon on the brane becomes de Sitter in static coordinates. Studying the
Einstein equations perturbatively we find the effective gravity on the brane at
this level and derive the 4D gravitational constant. Considering a homogeneous
and isotropic fluid in the corresponding FLRW coordinates we find that the bulk
fluid density below the brane, which has the same equation of state as the
fluid on the brane, contributes to the energy density in the Friedmann equation
and therefore in late time may be attributed to dark matter. Studying the
stability of the setup we observe that the brane becomes stabilized, in the
presence of matter on the brane, with a de Sitter length that is qualitatively
of the order of Schwarzschild radius of the universe due to matter. We briefly
discuss effects that can bound the de Sitter lifetime. In particular, this
model can provide a lifetime compatible with Trans-Planckian Censorship
conjecture for the current de Sitter phase.
