The microcalorimetric hydrogen titration technique combined with conventional volumetric measurements has been used to reveal peculiarities of the hydrogenation of the single-phase TiZrVNbTa equiatomic high-entropy alloy. The alloy has been produced in the form of microfibers by the pendent drop melt extraction technique. Palladium coating of the fibers has been applied to enable first hydrogenation at room temperature without additional activation. An analysis of the obtained data allows us to evaluate the dependence of hydrogenation enthalpy on the hydrogen concentration in the alloy. Three concentration ranges, presumably related to the formation of the hydrogen solid solution, monohydride and dihydride phases, have been identified, and the corresponding ΔH values of about −100, −80 and −60 kJ/mol H2, respectively, have been determined.