The recent measurements of the cosmological parameter $H_0$ from the direct local observations and the inferred value from the Cosmic Microwave Background show $sim 4 sigma$ discrepancy. This may indicate new physics beyond the standard $Lambda$CDM. We investigate the keV gravitino dark matter that has a small fraction of non-thermal component (e.g. from the late decay of NLSP bino) under various cosmological constraints. We find such a scenario is highly predictive and can be tested by searching for the dilepton plus missing energy events at the LHC. Besides, we also discuss its implication for Hubble tension, however, which can be reduced to $3sigma$ level marginally.
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