First, consider what apodization is. It is the intentional gradual vignetting of a lens, with the intention, typically, of producing better bokeh.

Think of it like a radial graduated ND filter being put in the aperture stop. The edge of the aperture might be almost black (non-transmissive) while the center is completely clear.

Because it's the aperture stop, everywhere in the field of view interacts equally with this effect.

As you close the aperture, the most truncated part is removed; in the limit of completely closed (say, f/22), the lens will be just like a regular camera lens.

What happens in between, at say f/8?

Well, you get something partway between a fully apodized and an unapodized lens.

How does this impact the landscape performance of the lens?

Well, in the diffraction limited case, an apodized lens has more contrast than an unapodized one, in exchange for reduced resolution. Depending what the f/# is, the level of detail that is lost may fall outside the capture range of your camera.

We can also assume that most lenses of high quality are in a diffraction limited regime at f/8 and smaller.

At these apertures, you get the "partially apodized" effect, and you will get a small increase in contrast in exchange for a small loss of resolution. The loss of resolution, again, may be at higher frequencies than your sensor can capture.

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I can show you a simulation if you want.