  Complex systems are often inherently non-ergodic and non-Markovian for which
Shannon entropy loses its applicability. In particular accelerating,
path-dependent, and aging random walks offer an intuitive picture for these
non-ergodic and non-Markovian systems. It was shown that the entropy of
non-ergodic systems can still be derived from three of the Shannon-Khinchin
axioms, and by violating the fourth -- the so-called composition axiom. The
corresponding entropy is of the form $S_{c,d} \sim \sum_i \Gamma(1+d,1-c\ln
p_i)$ and depends on two system-specific scaling exponents, $c$ and $d$. This
entropy contains many recently proposed entropy functionals as special cases,
including Shannon and Tsallis entropy. It was shown that this entropy is
relevant for a special class of non-Markovian random walks. In this work we
generalize these walks to a much wider class of stochastic systems that can be
characterized as `aging' systems. These are systems whose transition rates
between states are path- and time-dependent. We show that for particular aging
walks $S_{c,d}$ is again the correct extensive entropy. Before the central part
of the paper we review the concept of $(c,d)$-entropy in a self-contained way.
