In an earlier work, we proved a generalized entropy ergodic theorem forfinite nonhomogeneous Markov chains (NMC). In this paper, we establish a generalized strong law of large numbers for finite $m$-th order NMC. Then we deduce a generalized entropy ergodic theorem for finite $m$-th order NMC, under some assumptions on the continuity rate and of non-nullness. Explicit upper and lower bounds relating the generalized relative entropy density of the original finite non-null stationary sequence and its canonical $m$-order Markov approximation is obtained.
