  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 $\mstau \gtrsim 1 \unit{TeV}$ and puts an upper bound on
the reheating temperature $T_R \simeq 10^7\unit{GeV}$ which does not satisfy
the requirements for thermal leptogenesis. For the non-standard cosmological
scenario, the reheating temperature bound can be strongly relaxed $T_R\gg 10^9
\unit{GeV}$ and the lithium-7 problem solved with a stau typical mass of
$\mstau \sim 600-700 \unit{GeV}$ and down to $\sim 400 \unit{GeV}$ with a very
important dark component that could enable possible production and detection at
the LHC.
