  We explore the cosmological implications provided by the geodetic brane
gravity action corrected by an extrinsic curvature brane term, describing a
codimension-1 brane embedded in a 5D fixed Minkowski spacetime. In the geodetic
brane gravity action, we accommodate the correction term through a linear term
in the extrinsic curvature swept out by the brane. We study the resulting
geodetic-type equation of motion. Within a Friedmann-Robertson-Walker metric,
we obtain a generalized Friedmann equation describing the associated
cosmological evolution. We observe that, when the radiation-like energy
contribution from the extra dimension is vanishing, this effective model leads
to a self-(non-self)-accelerated expansion of the brane-like universe in
dependence on the nature of the concomitant \beta-parameter associated with the
correction, which resembles an analogous behaviour in the DGP brane cosmology.
Several possibilities in the description for the cosmic evolution of this model
are embodied and characterized by the involved density parameters related in
turn to the cosmological constant, the geometry characterizing the model, the
introduced \beta-parameter as well as the dark like-energy and the matter
content on the brane.
