There are two factors determined by physics that favor autofocusing with wider apertures.
- More light An f/2.8 lens lets in twice as much light as an f/4 lens. The more light an AF system has to work with the faster and more accurate it can be.
- Wider baseline Phase detection AF works by comparing the differences between the light coming from the right and left sides of the lens. The wider the effective aperture of the lens (more properly called the entrance pupil), the further apart the light rays that are compared can be.
To take advantage of the wider aperture, the pairs of sensors for a particular focus point in the AF array must be further apart from one another. But that makes those lines not very useful when a lens with a narrower aperture is attached to the lens. So camera manufacturers hedge their bet a little. Some of the focus points are more sensitive/accurate but only function well with a large aperture lens. Other focus points are tuned to be able to use the light from lenses with narrower apertures. But those points can't take advantage of the wider light rays provided by a wide aperture lens.
This is because the two lines on the focus array for each focus point are in a fixed position. If they are close enough to each other to be able to use light that gets through each side of the lens with a narrow f/8 aperture, they are not far enough apart from each other to sense the light that gets through the edge of the lens with a wide f/2.8 or wider aperture. Even when a faster lens is on the camera they are only using light falling on each side of the lens that is close enough to the center to make it through the narrower aperture.
How well the 70-200 f/4 takes advantage of your camera's focus system depends on the specific design parameters of you camera's focus system. In general though, a constant aperture f/4 telephoto lens will perform well. The only place you might be concerned is if you plan to use a tele-converter, since a 1.4X will raise the lens' maximum f-number to 5.6 and a 2X tele-converter will raise it to f/8.
Incidentally, the same physics that favors AF with lenses with wider apertures also favors DSLR cameras with larger sensors. Because the mirror is larger (particularly because it is wider) in a full frame camera than in an APS-C camera, the baseline used for the most sensitive focus points can also be wider.
For a little deeper answer on how cross type points work and a visualization of how f/2.8 points require lines that are further apart, see this answer.