  We generalize the theory of the inhomogeneity effect to enable comparison
among different inhomogeneous planets. A metric of inhomogeneity based on the
cumulative distribution function is applied to investigate the dependence of
planetary cooling on previously overlooked parameters. The mean surface
temperature of airless planets increases with rotational rate and surface
thermal inertia, which bounds the value in the tidally locked configuration and
the equilibrium temperature. Using an analytical model, we demonstrate that the
internal heat flux of giant planets exhibits significant spatial variability,
primarily emitted from the nightside and high-latitude regions acting as
``radiator fins." Given a horizontally uniform interior temperature in the
convective zone, the outgoing internal flux increases up to several folds as
the inhomogeneity of the incoming stellar flux increases. The enhancement
decreases with increasing heat redistribution through planetary dynamics or
rotation. The outgoing internal flux on rapidly rotating planets generally
increases with planetary obliquity and orbital eccentricity. The radiative
timescale and true anomaly of the vernal equinox also play significant roles.
If the radiative timescale is long, the outgoing internal flux shows a slightly
decreasing but nonlinear trend with obliquity. Our findings indicate that
rotational and orbital states greatly influence the cooling of planets and
impact the interior evolution of giant planets, particularly for tidally locked
planets and planets with high eccentricity and obliquity (such as Uranus), as
well as the spatial and temporal variations of their cooling fluxes.
