A class of entropic generating functionals is introduced to account for the stationary probability distributions arising in H theory, a framework for describing hierarchical multiscale complex systems. These entropies are indexed by the number $N$ of relevant scales in the hierarchy and by their corresponding universality classes, which are determined by the asymptotic behavior of the distribution tails, namely stretched-exponential or power-law forms. The associated stationary distributions are obtained by maximizing the corresponding functional under standard constraints, consistently with a canonical-ensemble description of a system in contact with a thermal reservoir. It is further shown that this general construction recovers the Tsallis entropy as a particular case for $N=1$ within the power-law class of distributions, thereby providing a natural interpretation of entropy nonextensivity in terms of the underlying hierarchical structure.