We present supersymmetric scenarios with gravitino LSP and stau NLSP in the case of a non-standard model of cosmology with the addition of a dark component in the pre-BBN era. In the context of the standard model of cosmology, gravitino LSP has drawn quite some attention as it is a good candidate for dark matter. It is produced in scattering processes during reheating after inflation and from the decay of the stau. With a long lifetime, the stau decays during Big Bang Nucleosynthesis. It is strongly constrained by the abundance of light elements but can however address the known ``BBN lithium problem''. It requires fairly massive staus m≳ 1TeV and puts an upper bound on the reheating temperature T_R ≃ 10⁷ GeV which does not satisfy the requirements for thermal leptogenesis. For the non-standard cosmological scenario, the reheating temperature bound can be strongly relaxed T_R ≫ 10⁹GeV and the lithium-7 problem solved with a stau typical mass of m ∼ 600–700 GeV and down to ∼ 400GeV with a very important dark component that could enable possible production and detection at the LHC.