  Non-linear cosmic structures contain valuable information on the expansion
history of the background space-time, the nature of dark matter, and the
gravitational interaction. The recently developed kinetic field theory of
cosmic structure formation (KFT) allows to accurately calculate the non-linear
power spectrum of cosmic density fluctuations up to wave numbers of
$k\lesssim10\,h\,\mathrm{Mpc}^{-1}$ at redshift zero. Cosmology and gravity
enter this calculation via two functions, viz. the background expansion
function and possibly a time-dependent modification of the gravitational
coupling strength.
  The success of the cosmological standard model based on general relativity
suggests that cosmological models in generalized theories of gravity should
have observable effects differing only weakly from those in standard cosmology.
Based on this assumption, we derive the functional, first-order Taylor
expansion of the non-linear power spectrum of cosmic density fluctuations
obtained from the mean-field approximation in KFT in terms of the expansion
function and the gravitational coupling strength. This allows us to study
non-linear power spectra expected in large classes of generalized gravity
theories. To give one example, we apply our formalism to generalized Proca
theories.
