Topological entropy of skew product map $F(x,y)=(f(x),g_x(y))$ in case of identical topological entropy for each fibre map $g_x$

Question

Suppose, we have a dynamical systen $(X,f)$ and a skew product $(X\times Y)$ with skew product map $$ F(x,y)=(f(x),g_x(y)) $$ with $g_x\colon Y\to Y$ for fixed $x\in X$.

I have a questions concerning topological entropy $h(F)$ of $F$:

Do we have that $$ h(F)=h(f)+c\qquad\text{ if } h(g_x)=c\text{ for all }x\in X? $$

(For example, if $h(g_x)=0$ for all $x\in X$, do we then have $h(F)=h(f)$?)