6. Given the function
$$
f(x)=A \cos \left(\omega x+\frac{\pi}{4} \omega\right)(A>0)
$$

is decreasing on $\left(0, \frac{\pi}{8}\right)$. Then the maximum value of $\omega$ is
Ni,6.8.
Assume $\omega>0$.
To make $f(x)$ a decreasing function in $\left(0, \frac{\pi}{8}\right)$, combining the image of the cosine-type function, we must have
$$
\begin{array}{l}
\frac{T}{2} \geqslant \frac{\pi}{8} \Rightarrow \frac{\pi}{\omega} \geqslant \frac{\pi}{8} \Rightarrow \omega \leqslant 8 . \\
\text { When } \omega=8 \text {, } \\
f(x)=A \cos (8 x+2 \pi)(A>0),
\end{array}
$$

Obviously, in $\left(0, \frac{\pi}{8}\right)$, it is a decreasing function, which fits.
Therefore, the maximum value of $\omega$ is 8.