I messed up my last question here. Should have contained all cases for $p$.
Does the Frechet Differential of the Operator $F: L^p([0,1]) \to L^p([0,1]),~~f \to \cos f$ at $f \equiv 0$ exist?
My candiate for $F'$ would be $h\sin f$, so
$$ \lim_{h \to0 } \frac{\|\cos h - 1-0\|_{L^p([0,1])}}{\|h\|_{L^p([0,1])}}$$
But I am not sure how to handle the $L^p$ Norm here, it seems completly different to $L^{\infty}$ Norm. I dont think that for every $h_k \to 0$, $\cos h_k -1$ goes to $0$ but I dont find any suitable example. I am thankful for hints.
No, this is not Frechet differentiable for all $p<\infty$. Take $h=\chi_{[0,1/n]}2\pi$. Set $f=0$. Then $$ F(f+h)-F(f)=F(h)-F(0)=0. $$ If $F$ would be Frechet differentiable, then this implies $F'(0)=0$. But $$ F(h/2)-F(0)-F'(0)(h/2) = -2\cdot \chi_{[0,1/n]} = h/\pi, $$ which is not $o(h)$.