  Subshifts of deterministic substitutions are ubiquitous objects in dynamical
systems and aperiodic order (the mathematical theory of quasicrystals). Two of
their most striking features are that they have low complexity (zero
topological entropy) and are uniquely ergodic. Random substitutions are a
generalisation of deterministic substitutions where the substituted image of a
letter is determined by a Markov process. In stark contrast to their
deterministic counterparts, subshifts of random substitutions often have
positive topological entropy, and support uncountably many ergodic measures.
The underlying Markov process singles out one of the ergodic measures, called
the frequency measure. Here, we develop new techniques for computing and
studying the entropy of these frequency measures. As an application of our
results, we obtain closed form formulas for the entropy of frequency measures
for a wide range of random substitution subshifts and show that in many cases
there exists a frequency measure of maximal entropy. Further, for a class of
random substitution subshifts, we prove that this measure is the unique measure
of maximal entropy. These subshifts do not satisfy Bowen's specification
property or the weaker specification property of Climenhaga and Thompson and
hence provide an interesting new class of intrinsically ergodic subshifts.
