Geopotential Height Clarification

[J Trueblood]

Hello everyone,
I am currently reading through Holton's An Introduction to Dynamic Meteorology. The whole thing is just a bit over my head, but I'm hoping to learn through immersion..Anyways, I'm kind of getting confused about the whole geopotential height definition.
If I'm correct, geopotential at a certain height (in what units?) is the work requires to raise a unit mass to height z from sea level. So basically the integral of gravity form sea level to height z.

Then geopotential height is geopotential divided by gravity, again...That being said..How is it possible that geopoential height varies at a given pressure? I don't see how it could vary if it only depends on height (which we make constant on say a 500mb map) and gravity (which is obviously a constant)...Can someone help me out here by explaining geopotential height? thank you

 

[J Trueblood]

Okay..I think I am starting to understand this. My mistake is that I was looking at height in terms of meters, when in reality, geopotential is a function of constant pressure. Because pressure depends on temperature and density, it is possible for a pressure to be found a one height somewhere, and at a different height somewhere else? Therefore, when making a geopotential height map of say 500mb, we have different heights (in meters) inf certain areas..because that pressure could be higher or lower off the ground depending on its thermodynamic properties. Is this correct?