Obstructions to the uniform stability of a $C_0$-semigroup

Let $T_t\colon X\to X$ be a $C_0$-semigroup with generator $A$. We prove that if the abscissa of uniform boundedness of the resolvent $s_0(A)$ is greater than zero then for each nondecreasing function $h(s)\colon\mathbb R_+\to\mathbb R_+$ there are $x'\in X'$ and $x\in X$ satisfying $\int_0^\infty h(|\langle x',T_tx\rangle|)\,dt=\infty$. If $i\mathbb R\cap\operatorname{Sp}(A)\ne\varnothing$ then such $x$ may be taken in $D(A^\infty)$.