To simplify the expression \((z^3)^0 \cdot z^5\), we can start by applying the property of exponents that states any non-zero number raised to the power of 0 is 1. Therefore, \((z^3)^0 = 1\).
Now substituting this into the expression, we have:
\[ (z^3)^0 \cdot z^5 = 1 \cdot z^5 = z^5 \]
So, the simplified expression is:
\[ \boxed{z^5} \]