4. 11 .

From the given, we know that
$$
f(\pi+x)=f(-\pi+x)=-f(\pi-x) \text { . }
$$

Let $x=0$, we get $f(\pi)=0$.
On the other hand,
$$
f(2 \pi-4)=f(-4)=-f(4)=0 \text { . }
$$

Similarly, $f(2 \pi-3)=0$.
Therefore, the zeros of $f(x)$ in the interval $[0,10]$ must include
$$
\begin{array}{l}
0, \pi, 2 \pi, 3 \pi, 3, 2 \pi+3, 2 \pi-3, 4 \pi-3, 4, \\
2 \pi-4, 4 \pi-4 .
\end{array}
$$

It is easy to construct $f(x)$ such that it has exactly these 11 zeros.
4. Let $f(x)$ be an odd function with domain $\mathbf{R}$ and period $2 \pi$, and $f(3)=f(4)=0$. Then $f(x)$ has at least $\qquad$ zeros in the interval $[0,10]$.

