We investigate the so-called superWIMP scenario with the gravitino as the lightest supersymmetric particle (LSP) in the context of nonstandard cosmology, in particular, brane world cosmology. As a candidate of the next-to-LSP (NLSP), we examine the slepton and the sneutrino. Brane world cosmological effects dramatically enhance the relic density of the slepton or sneutrino NLSP, so that the NLSP with mass of order 100 GeV can provide the correct abundance of gravitino dark matter through its decay. We find that with an appropriate five-dimensional Planck mass, this scenario can be realized consistently with the constraints from big bang nucleosynthesis for both NLSP candidates of the slepton and the sneutrino. The big bang nucleosynthesis constraints for the slepton NLSP are more stringent than that for the sneutrino; as the result, the gravitino must be rather warm in the slepton NLSP case. The energy density of the gravitino produced by thermal scattering is highly suppressed and negligible due to the brane world cosmological effects.