For the general case you can define a function that assigns to each point on the sphere the distance (angle) to the nearest boundary.
The desired number is then the integral of this function over the whole sphere (see example by Joriki).
For the sphere, we can consider, for example, circular regions (a hemisphere is a special case).
Inside a small circular region (cap) the distance function looks like a little cone, centered at the center of the circle.
If the black (and therefore also the white) regions are made up of unions, intersection or differences of caps the distance function can be constructed from the distance functions for each of the caps.
Note that the colors are irrelevant.
The only thing that matters is the distance to the nearest boundary.
All you need is the list of circles.
Performing the integration exactly would not be easy.
The case of two non-overlapping caps is doable (non-intersecting circles).
For a set of circles that is not too large, you could use Monte Carlo integration (for a random set of points, average the distance to the nearest circle).
Easy to implement, but not very efficient.
Monte Carlo would also work for more general cases.
You could consider intersections of hemispheres as the equivalent of polygons on a sphere.
So the ideas described above apply to the case of "polygonal" regions on the sphere.
If you approximate your boundary by a discrete set of points you can probably use a Voronoi diagram.
Note that this kind of question is much easier to think about on a unit square (with periodic boundary conditions) or easier still on a unit interval (with periodic boundary conditions).
In the latter case, the distance function for an arbitrary partition is made up of "hats" with slopes of 1 and -1.
The integration is easy to perform in this case.