5. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, a sphere is constructed with vertex $A$ as the center and $\frac{2 \sqrt{3}}{3}$ as the radius. The length of the curve formed by the intersection of the sphere's surface with the cube's surface is $\qquad$
5. $\frac{5 \sqrt{3} \pi}{6}$.

The sphere intersects with all six faces of the cube, resulting in two types of intersection lines: one type on the three faces containing vertex $A$, i.e., faces $A A_{1} B_{1} B$, $A B C D$, and $A A_{1} D_{1} D$; the other type on the three faces not containing vertex $A$, i.e., faces $B B_{1} C_{1} C$, $C C_{1} D_{1} D$, and $A_{1} B_{1} C_{1} D_{1}$.
(1) On face $A A_{1} B_{1} B$, the intersection line is arc $\overparen{E F}$ and lies on the great circle passing through the center of the sphere $A$.
Given $A E=\frac{2 \sqrt{3}}{3}, A A_{1}=1$
$\Rightarrow \angle A_{1} A E=\frac{\pi}{6}$.
Similarly, $\angle B A F=\frac{\pi}{6}$.
Thus, $\angle E A F=\frac{\pi}{6}$.
Therefore, the length of arc $\overparen{E F}$ is $\frac{2 \sqrt{3}}{3} \cdot \frac{\pi}{6}=\frac{\sqrt{3} \pi}{9}$, and there are three such arcs.
(2) On face $B B_{1} C_{1} C$, the intersection line is arc $\overparen{F G}$ and lies on the small circle formed by the intersection of the plane at a distance of 1 from the center of the sphere with the sphere, with the center of the small circle at $B$ and radius $\frac{\sqrt{3}}{3}, \angle F B G=\frac{\pi}{2}$, so the length of arc $\overparen{F G}$ is $\frac{\sqrt{3}}{3} \cdot \frac{\pi}{2}=\frac{\sqrt{3} \pi}{6}$, and there are three such arcs.
In summary, the total length of the curves is
$$
3 \times \frac{\sqrt{3} \pi}{9}+3 \times \frac{\sqrt{3} \pi}{6}=\frac{5 \sqrt{3} \pi}{6} .
$$