12. The base of a hexagonal prism is a regular hexagon, and its lateral edges are perpendicular to the base. It is known that all vertices of the hexagonal prism lie on the same sphere, and the volume of the hexagonal prism is $\frac{9}{8}$, with the perimeter of the base being 3. Then the volume of this sphere is $\qquad$
12. $\frac{4}{3} \pi$.

Since the base circumference is 3, the base side length is $\frac{1}{2}$, and the base area is $S=\frac{3}{8} \sqrt{3}$.

Given the volume is $\frac{9}{8}$, the height is $\sqrt{3}$. The diameter of the sphere is $\sqrt{1^{2}+(\sqrt{3})^{2}}=2$, and the volume of the sphere $V=\frac{4}{3} \pi R^{3}=\frac{4}{3} \pi$.