10. Let the function be
$$
f(x)=\cos 2 x+2 \sqrt{3} \sin x \cdot \cos x(x \in \mathbf{R}) \text {. }
$$

If there are ten distinct positive numbers $x_{i}$ satisfying
$$
f\left(x_{i}\right)=2, x_{i}<10 \pi(i=1,2, \cdots, 10),
$$

then $x_{1}+x_{2}+\cdots+x_{10}=$ . $\qquad$
10. $\frac{140 \pi}{3}$.

From the problem, we have $f(x)=2 \sin \left(2 x+\frac{\pi}{6}\right)$.
Also, $f\left(x_{i}\right)=2$, thus,
$$
\begin{array}{l}
\sin \left(2 x_{i}+\frac{\pi}{6}\right)=1 \\
\Rightarrow 2 x_{i}+\frac{\pi}{6}=2 k \pi+\frac{\pi}{2} \\
\Rightarrow x_{i}=k \pi+\frac{\pi}{6}(k \in \mathbf{Z}) .
\end{array}
$$

Since $0<x_{i}<10 \pi$, we have $k=0,1, \cdots, 9$.
Therefore, $x_{1}+x_{2}+\cdots+x_{10}$
$$
\begin{array}{l}
=(1+2+\cdots+9) \pi+10 \times \frac{\pi}{6} \\
=\frac{140 \pi}{3}
\end{array}
$$