For any $t \in \mathbb{R}$ the substitutions $x = t $ and $x =\sqrt[3]{1-t^3}$ give the following system linear in $f(t)$ and $f(\sqrt[3]{1-t^3})$: $$f(t) + 2f(\sqrt[3]{1-t^3}) = t^3,$$$$f(\sqrt[3]{1-t^3})+2f(t) = 1-t^3,$$which gives the unique solution $f \equiv \frac{2}{3} - x^3$, which clearly satisfies the given equation.