  A principle of hierarchical entropy maximization is proposed for generalized
superstatistical systems, which are characterized by the existence of three
levels of dynamics. If a generalized superstatistical system comprises a set of
superstatistical subsystems, each made up of a set of cells, then the
Boltzmann-Gibbs-Shannon entropy should be maximized first for each cell, second
for each subsystem, and finally for the whole system. Hierarchical entropy
maximization naturally reflects the sufficient time-scale separation between
different dynamical levels and allows one to find the distribution of both the
intensive parameter and the control parameter for the corresponding
superstatistics. The hierarchical maximum entropy principle is applied to
fluctuations of the photon Bose-Einstein condensate in a dye microcavity. This
principle provides an alternative to the master equation approach recently
applied to this problem. The possibility of constructing generalized
superstatistics based on a statistics different from the Boltzmann-Gibbs
statistics is pointed out.
