  We obtain an analog of the prime number theorem for a class of branched
covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are
topological models of some non-uniformly expanding rational maps without any
smoothness or holomorphicity assumptions. More precisely, we show that the
number of primitive periodic orbits, ordered by a weight on each point induced
by an (eventually) positive real-valued H\"{o}lder continuous function on $S^2$
that is not cohomologous to a constant, is asymptotically the same as the
well-known logarithmic integral. In particular, our results apply to
postcritically-finite rational maps for which the Julia set is the whole
Riemann sphere.
