Dark matter constitutes the major contribution to the matter content of the universe, yet its true nature is still a mystery today. A vast and diverse experimental effort, especially dedicated to weakly interacting massive particles (WIMP), has not yet found conclusive DM evidence. This has reinvigorated interest in alternative dark matter candidates including more feebly coupled particles sometimes called FIMPs. Despite their typically very suppressed coupling to ordinary matter, it is still possible to probe FIMPs nature with particle physics experiments and Cosmology. In my talk, I will first discuss this interesting interplay and then focus on not even feebly coupled dark matter which could be produced in the early universe from primordial black holes evaporation simply due to their gravitational interactions. The common feature of the dark matter models discussed is their possibly testable “warm dark matter-like” cosmological imprint, or more generally “non-cold dark matter” imprint.