  Gravitino Dark Matter represents a compelling scenario in Supersymmetry,
which brings together a variety of data from cosmology and collider physics. We
discuss the constraints obtained from the LHC on supersymmetric models with
gravitino dark matter and neutralino NLSP, which is the case most difficult to
disentangle at colliders from a neutralino LSP forming DM. The phenomenological
SUSY model with 19+1 free parameters is adopted. Results are obtained from
broad scans of the phase space of these uncorrelated parameters. The relation
between gravitino mass, gluino mass and reheating temperature as well as the
derived constraints on these parameters are discussed in detail. This relation
offers a unique opportunity to place stringent bounds on the cosmological
model, within the gravitino dark matter scenario, from the results of the LHC
searches in Run-2 and the planned High-Luminosity upgrade.
