In addition to the answer by @Kaberett lapse rate explains the height variation of the tropopause quite well. Lapse rate is a property of gravity and the ideal gas law and it has to be adjusted for circulation and condensation and the top-down heating from the sun, which is a small effect given a mostly transparent atmosphere. Lapse rate isn't perfect, but it's still good for an approximation.
Shorter, Britannica summary
From the Britannica summary:
The lapse rate of nonrising air—commonly referred to as the normal, or
environmental, lapse rate—is highly variable, being affected by
radiation, convection, and condensation; it averages about 6.5 °C per
kilometre (18.8 °F per mile) in the lower atmosphere (troposphere).
If we take that averaged 6.5°C per km, that explains pretty close to the surface temperature variation on the Earth at sea level. (Antarctica, being mostly more than a mile in elevation, you need to adjust the temperature -6.5°C per km in altitude for surface temperature). The troposphere being an average of 8 km higher at the equator, corresponds to 52°c variation using that -6.5 degree estimate, or about 32 degrees (equator) to minus 20 (pole), which is very much in the ballpark.
Also, as noted in the Wikipedia article
the dry adiabatic lapse rate is a constant 9.8 °C/km (5.38 °F per
1,000 ft, 3 °C/1,000 ft), the moist adiabatic lapse rate varies
strongly with temperature. A typical value is around 5 °C/km
Moist air has a lower lapse rate, which explains why the troposphere is highest over warm equatorial Pacific and why it peaks over monsoons. - see this article.
It's the surface temperature and water vapor. That's the gist of it. The rotation of the Earth can't account for a troposphere that's twice as high over the equator. The rotation of the Earth (and if we want to get silly, also the Moon's atmospheric tidal bulges) both affect the troposphere some, but they are minor players.
Earth's atmosphere shouldn't bulge much more than it's body, about 1 part in 298, even if you multiply that by 4, and I think that's too aggressive a mathematical approach, but even so, that would be less than 1 part in 70 of atmospheric expansion over the equator due to the spin of the Earth. That's very far from the doubling that temperature explains.