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We obtain an analog of the prime number theorem for a class of branchedcovering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which aretopological models of some non-uniformly expanding rational maps without anysmoothness or holomorphicity assumption. More precisely, we show that thenumber of primitive periodic orbits, ordered by a weight on each point inducedby a non-constant (eventually) positive real-valued H\"{o}lder continuousfunction on $S^2$ satisfying some additional regularity conditions, isasymptotically the same as the well-known logarithmic integral, with anexponential error term. In particular, our results apply topostcritically-finite rational maps for which the Julia set is the wholeRiemann sphere. Moreover, a stronger result is obtained for Latt\`{e}s maps.

Prime orbit theorems for expanding Thurston maps: Latt\`es maps and split Ruelle operators