I use thermal convection models to search for combinations of physical parameters that are compatible with the results of xenolith studies on the history and present thermal structure of cratonal lithosphere. The cratonal lithosphere above ~180 km depth formed in the Archean and remained stable until recently sampled. The mantle adiabat cooled ~150 K over this time. The temperature change across the rheologically active boundary layer at the lithospheric base is <300 K over a depth range of several tens of kilometers. Modern cratonal lithospheric thicknesses are relatively uniform, ~225 km, much thicker than old oceanic lithosphere. Cratonal lithosphere is now in quasi-steady state with conductive heat flow to the surface in balance with heat supplied to the base of the lithosphere by convection driven by local temperature contrasts within the rheologically active boundary layer. This heat flow and the lithosphere thickness changed little after the cratonal lithosphere stabilized. One possibility is that chemically buoyant lithosphere forms a conductive lid above the convecting normal mantle. To survive, the chemical lithosphere also needs to be more viscous than normal mantle. A reasonable situation has chemical lithosphere a factor of 20 more viscous than normal mantle with weakly temperature-dependent viscosity (a factor of e over 100 K) from 0.2 ¿ 1020 Pa s along the modern mantle adiabat. However, a chemical lid is unnecessary for lithospheric thickness to change slowly over time. For example, the base of the lithosphere may be stable if the viscosity at its base (along an adiabat) decreases rapidly with depth.