Stagnant-lid convection, where subduction and surface plate motion is absent, is common among the rocky planets and moons in our solar system, and likely among rocky exoplanets as well. How stagnant-lid planets thermally evolve is an important issue, dictating not just their interior evolution but also the evolution of their atmospheres via volcanic degassing. On stagnant-lid planets, the crust is not recycled by subduction and can potentially grow thick enough to significantly impact convection beneath the stagnant lid. We perform numerical models of stagnant-lid convection to determine new scaling laws for convective heat flux that specifically account for the presence of a buoyant crustal layer. We systematically vary the crustal layer thickness, crustal layer density, Rayleigh number and Frank-Kamenetskii parameter for viscosity to map out system behaviour and determine the new scaling laws. We find two end-member regimes of behaviour: a 'thin crust limit', where convection is largely unaffected by the presence of the crust, and the thickness of the lithosphere is approximately the same as it would be if the crust were absent; and a 'thick crust limit', where the crustal thickness itself determines the lithospheric thickness and heat flux. Scaling laws for both limits are developed and fit the numerical model results well. Applying these scaling laws to rocky stagnant-lid planets, we find that the crustal thickness needed for convection to enter the thick crust limit decreases with increasing mantle temperature and decreasing mantle reference viscosity. Moreover, if crustal thickness is limited by the formation of dense eclogite, and foundering of this dense lower crust, then smaller planets are more likely to enter the thick crust limit because their crusts can grow thicker before reaching the pressure where eclogite forms. When convection is in the thick crust limit, mantle heat flux is suppressed. As a result, mantle temperatures can be elevated by 100 s of degrees K for up to a few Gyr in comparison to a planet with a thin crust. Whether convection enters the thick crust limit during a planet's thermal evolution also depends on the initial mantle temperature, so a thick, buoyant crust additionally acts to preserve the influence of initial conditions on stagnant-lid planets for far longer than previous thermal evolution models, which ignore the effects of a thick crust, have found.
