  Weak gravitational lensing convergence peaks, the local maxima in weak
lensing convergence maps, have been shown to contain valuable cosmological
information complementary to commonly used two-point statistics. To exploit the
full power of weak lensing for cosmology, we must model baryonic feedback
processes because these reshape the matter distribution on non-linear and
mildly non-linear scales. We study the impact of baryonic physics on the number
density of weak lensing peaks using the FLAMINGO cosmological hydrodynamical
simulation suite. We generate ray-traced full-sky convergence maps mimicking
the characteristics of a Stage IV weak lensing survey. We compare the number
densities of peaks in simulations that have been calibrated to reproduce the
observed galaxy mass function and cluster gas fraction or to match a shifted
version of these, and that use either thermally driven or jet AGN feedback. We
show that the differences induced by realistic baryonic feedback prescriptions
(typically $5 - 30\%$ for $\kappa = 0.1 - 0.4$) are smaller than those induced
by reasonable variations in cosmological parameters ($20 - 60\%$ for $\kappa =
0.1 - 0.4$) but must be modeled carefully to obtain unbiased results. The
reasons behind these differences can be understood by considering the impact of
feedback on halo masses, or by considering the impact of different cosmological
parameters on the halo mass function. Our analysis demonstrates that, for the
range of models we investigated, the baryonic suppression is insensitive to
changes in cosmology up to $\kappa \approx 0.4$ and that the higher $\kappa$
regime is dominated by Poisson noise and cosmic variance.
