Motivated by the absence of signals of new physics in searches for both new particles at the LHC and a weakly interacting massive particle dark matter candidate, we consider a scenario where supersymmetry is broken at a scale above the reheating temperature. The low-energy particle content then only consists of Standard Model states and a gravitino. We investigate the possibility that the latter provides the main component of dark matter through the annihilation of thermalized Standard Model particles. We focus on the case where its production through scattering in the thermal plasma is well approximated by the nonlinear supersymmetric effective Lagrangian of the associated Goldstino and identify the parameter space allowed by the cosmological constraints, allowing the possibility of a large reheating temperature compatible with leptogenesis scenarios and alleviating the so-called " gravitino problem " .