How to input thickness in a spherical gyroid

Question

I have the equation to contourplot3D a spherical gyroid given below

r = 2 Pi;
ContourPlot3D[
 Sin[x] Cos[y] + Sin[y] Cos[z] + Sin[z] Cos[x] == 0,
 {x, -r, r}, {y, -r, r}, {z, -r, r},
 RegionFunction -> ({x, y, z} \[Function] x^2 + y^2 + z^2 <= r^2),
 Mesh -> None
 ]

When I run the code I can not obtain the volume. i will like to know how i can input a thickness in the above equation. There are some parameters like the Volume, thickness and the specific surface area which i need to determine