  A novel formalism, called H-theory, is applied to the problem of statistical
equilibrium of a hierarchical complex system with multiple time and length
scales. In this approach, the system is formally treated as being composed of a
small subsystem---representing the region where the measurements are made---in
contact with a set of `nested heat reservoirs' corresponding to the
hierarchical structure of the system. The probability distribution function
(pdf) of the fluctuating temperatures at each reservoir, conditioned on the
temperature of the reservoir above it, is determined from a maximum entropy
principle subject to appropriate constraints that describe the thermal
equilibrium properties of the system. The marginal temperature distribution of
the innermost reservoir is obtained by integrating over the conditional
distributions of all larger scales, and the resulting pdf is written in
analytical form in terms of certain special transcendental functions, known as
the Fox $H$-functions. The distribution of states of the small subsystem is
then computed by averaging the quasi-equilibrium Boltzmann distribution over
the temperature of the innermost reservoir. This distribution can also be
written in terms of $H$-functions. The general family of distributions reported
here recovers, as particular cases, the stationary distributions recently
obtained by Mac\^edo {\it et al.} [Phys.~Rev.~E {\bf 95}, 032315 (2017)] from a
stochastic dynamical approach to the problem.
