[steve in DC]

A finite amount of air (or steel, or water, or whatever) has a given mass, no matter what temperature it may be.

The earth,s atmosphere is finite. Thus its mass is the same, no matter what its density might be.

Gravity is said to be inexorably tied to mass, not density. Giant gas planets have a gravitational pull that correlates with their mass, not their density. Although, that is not necessarily correct, in that their mass has been back calculated based upon the strength of their gravitational pull.

[swilhelm73]

A finite amount of air (or steel, or water, or whatever) has a given mass, no matter what temperature it may be.

Density can be consider to be the measure of the amount of matter within a given area. The ambient temperature is the primary determinate of the density of air. Cold air is denser the warmer air. As such, there are more "air" molecules in the area around the pendulum if the air is cooled.

So, cold air does indeed generate a stronger gravitational pull then warmer air.

However, as I pointed out originally this effect would be very very small.