I know that the Klein-Gordon equation in general relativity takes the form (a massless field)
$\nabla_\mu \nabla^\mu \phi=\sum_{a,b} \frac{1}{\sqrt{-g}}\partial_a(\sqrt{-g}g^{ab}\partial_b\phi) =0$
and that the spectrum of energies is $\omega_p=\sqrt{p^2+m^2}$, is it the same spectrum for curved space-time?