When looking at the planetary data it seems as if the tropopause appears typically at a pressure of 100 mB and that the temperature at this reference pressure raised to the power of 4 is roughly proportional to the sum of the solar and internal heat forcings. It is primarily Jupiter and Saturn that has substantial internal heat sources. We could therefore test the following conjecture:
Recall the formula for a bouyant layer of air
where theta is a constant and p_0 is the reference pressure. Based on the observational evidence we put the reference pressure to
Finally the constant theta is given by
where F_s is the solar forcing, F_i is the internal forcing and k is the only "phenomenological" parameter in the model. If I remember correctly, in SI units it is roughly equal to 6e-7.
Obviously the temperature follows the pressure, but, one should be careful not to immediately infer that the pressure is the sole "cause" of the elevation in air temperature. It is merely one of the factors in a bigger picture. In a way one could say that the pressure puts a limit to the rate of convective cooling due to bouyancy.
