  Expanding Thurston maps form a class of branched covering maps on the
topological $2$-sphere $S^{2}$ which are topological models of some
non-uniformly expanding rational maps without any smoothness or holomorphicity
assumption initially investigated by W. P. Thurston, M. Bonk, D. Meyer, P.
Ha\"issinsky, and K. M. Pilgrim. The measures of maximal entropy and the
absolutely continuous invariant measures for these maps have been studied by
these authors, and equilibrium states by the first-named author. In this paper,
we initiate the investigation on two new classes of invariant measures (namely,
the maximizing measures and ground states), and establish the Liv\v{s}ic
theorem, a local Anosov closing lemma, and give a positive answer to the
Typically Periodic Optimization Conjecture from ergodic optimization for these
maps. As an application, we establish these results for Misiurewicz--Thurston
rational maps (i.e., postcritically-finite rational maps without periodic
critical points) on the Riemann sphere including the Latt\`es maps with respect
to the spherical metric. Our strategy relies on the visual metrics developed by
the above authors.
  In particular, we verify, in a first non-uniformly expanding setting, the
Typically Periodic Optimization Conjecture, establishing that for a generic
H\"{o}lder continuous potential, there exists a unique maximizing measure,
moreover, this measure is supported on a periodic orbit, it satisfies the
locking property, and it is the unique ground state. The expanding Thurston
maps we consider include those that are not topologically conjugate to rational
maps, in particular, they can have periodic critical points.
