Would an ideal gas be colder at higher altitude due to gravity?

Since gas molecules are affected by gravity, wouldn't that make gas molecules at higher than average elevation slower (at the top of their ballistic parabola) and thus colder than air molecules accelerating to the ground?

In non-relativistic theory no, because in thermodynamic equilibrium temperature has to be the same everywhere. The slowing down does not occur on average because the molecules do not move along ballistic parabola, but collide with each other (in very rarified gas the collisions are rare and the establishment of the same temperature as down below may take long time).

In relativistic theory, yes because in thermodynamic equilibrium places with lower gravitational potential should have higher temperature (https://en.wikipedia.org/wiki/Ehrenfest%E2%80%93Tolman_effect); but the predicted difference is negligibly small for common gravitational fields like Earth's.