"pV = nRT"

Alarmists and Physics Deniers don't actually know what this equation signifies and make themselves look silly by trotting it out as if it somehow supports Alarmist Theory and/or debunks the Gravito-Thermal Effect.

Nothing of the sort. There's a full explanation and worked example at ChemGuide.co.uk. This is basic first year level GCSE Physics and nothing controversial.
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Let's apply it to typical temp, pressure, density at sea level:
Pressure = 101,325 Pa
Volume = 1 m3
Mass of air in 1 m3 at 'standard temperature and pressure' = 1.227 kg
n = number of moles of gas in 1 m3 = mass/m3 divided by molecular mass of 'air', which is 29g/mole
R = universal gas constant* = 8.31441 J/K/mol
Temp = 288K

Stick in the numbers on the right hand side, 1,227/29 x 8.31441 x 288 = 101,314. Close enough to 101,325!
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Now we've got the hang of it, what's the likely temperature at 10 km altitude?
Wiki tells us that pressure up there is 26,300 Pa, density is 416 g/m3.
The left hand side is 26,300 x 1 = 26,300
The right hand side is 416/29 x 8.31441 x T = 119.3 x T
So 119.3 T = 26,300; and T = 26,300/119.3 = 220K

Which is exactly what the blue line on Wiki's chart - and real life measurement - show. Wiki's chart is what you get if you just start by assuming ever increasing density in a gravitational/acclerating field and working from there (see diagram 5).

In fact, you can assume constant density, desnity which increases linearly or geometrically as you go down, there would still be a similar profile with increasing pressure and hence temperature towards the surface. Basic maths. Any other outcome is mathematically impossible unless you assume that density increases with height at an implausible rate (in which case, what happens at the top?).
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* Keen-eyed readers will know that this is Avogadro's number x Boltzmann's constant.