  We consider Gaussian random waves on hyperbolic spaces and establish variance
asymptotics and central limit theorems for a large class of their integral
functionals, both in the high-frequency and large domain limits. Our strategy
of proof relies on a fine analysis of Wiener chaos expansions, which in turn
requires us to analytically assess the fluctuations of integrals involving
mixed moments of covariance kernels. Our results complement several recent
findings on non-linear transforms of planar and arithmetic random waves, as
well as of random spherical harmonics. In the particular case of 2-dimensional
hyperbolic spaces, our analysis reveals an intriguing discrepancy between the
high-frequency and large domain fluctuations of the so-called fourth
polyspectra -- a phenomenon that has no counterpart in the Euclidean setting.
We develop applications of a geometric flavor, most notably to excursion
volumes and occupation densities.
