First studies on the mechanical behavior of high-entropy alloys (HEAs) in high-pressure hydrogen environment are available. In contrast, the underlying hydrogen absorption, diffusion and trapping in these HEAs like the Cantor-alloy was less in the scientific scope so far. For that reason, the CoCrFeMnNi-HEA was compared to a conventional AISI 316L austenitic steel, by exposing to high-pressure hydrogen charging at 200 bar and very-high pressure at 1,000 bar. Thermal desorption analysis (TDA) was applied with different heating rates (0.125 K/s to 0.500 K/s). The underlying TDA spectra were analyzed in terms of a reasonable peak deconvolution to into a defined number of peaks and the calculation of the activation energies for the respective and predominant hydrogen trap sites. Both materials show a comparable hydrogen diffusivity. The obtained activation energies suggest that in case of CoCrFeMnNi-HEA an interaction of the austenitic phase as well as the direct atomic bonding of hydrogen to the metal atoms are the dominant traps. Available literature suggests that the Cr and Mn-content is here of special interest for the direct hydrogen bonding at solute atoms. Despite the activation energy, the trap occupancy rate must be considered in terms of a pressure-related hydrogen absorption. The derived apparent hydrogen solubility was in the order: 316L < CoCrFeMnNi-HEA for both charging pressures. Especially, the 1,000 bar values lead to noteworthy results with > 70 wt.ppm for the AISI 316L and >130 wt.ppm for the CoCrFeMnNi. In fact, both the hydrogen diffusion and trapping data on gaseous high-pressure hydrogen charged HEAs are rare so far. The results of the present study allow a deeper understanding of hydrogen trapping in the regarded CoCrFeMnNi-system.