Are you sure?
The physical aperture is the same, I agree. But the 'f stop' is the ratio of physical aperture size to focal length, so if you add a teleconverter to go from 400mm to 800mm, the physical aperture is constant so the f value changes accordingly.
That's what I wrote,
the physical aperture of the 300mm lens remains the same. The light passes through a 300mm lens at F5.6
before it reaches the converter. Imagine a completely manual lens with no automatic read-out of F/stop, and no automatic adjustment of F/stop. Imagine you have to change the aperture by twisting a ring on the lens.
You manually set the f/stop on the 300mm lens to F/5.6. You add a 2x converter. The 300mm lens still shows a reading of F5.6 on the barrel. However, if you then take a shot on the assumption that the two combined lenses are F5.6, the shot will be underexposed by 2 stops. This is because the 2x converter grabs one quarter of the image, or light, that has passed through the 300mm lens at F5.6 and spreads it over the entire sensor, so that each pixel receives only one quarter of the light that it would have received without the converter.
This also, perhaps, explains more clearly why the DoF does not change. The image that has passed through the converter is an F5.6 image with the DoF of an F5.6 image (in relation to the 300mm lens). Although the automatic f/stop readout on a modern camera changes to F11 when the converter is added, to ensure correct exposure, the qualities of that F5.6 image (what's sharp, what's not sharp, what's in focus, what's out-of-focus etc), remain unchanged, ideally, if the converter is a perfect lens.
But here's the problem. There's no such thing as a perfect lens. Even the best converter available will degrade that F5.6 image to some degree. However, such degradation is offset, at least partially if not completely, by the increased pixel count of the final image.
This is the same principle that applies to all new camera models boasting a higher pixel count, such as the recently announced 50mp Canon 5DS. The immediate reaction of some people is that they don't need so many megapixels and that they are unnecessary for their purposes and just slow down all processing. I wonder if such people are aware that increasing the pixel count of a sensor, whilst maintaining the basic pixel quality of previous models, has the effect of upgrading any lens that is used with the camera, whether the lens be high quality or mediocre.
As I've mentioned before, whatever the quality of the lens used with a converter, the new lens of longer focal length that results, will be a lower quality lens, period.