Let $X:=\{X_t\}_{t\in[0,T]}$ be the unique strong solution to the Itô diffusion $$ dX_t = a(t,X_t)dt + b(t,X_t)dW_t, $$ where $a,b$ are such that the conditions for the existence of the invariant measure are satisfied. How would one go about showing that for any $f\in\mathcal{C}(\mathbb{R})$: $$ \mathbb E[|f(X_t)-m|^p]^{1/p}\leq C \text{ uniformly in } t\in[0,T], $$ where $m$ is the expectation taken under the invariant measure $\mu$, i.e., $m=\mathbb E^\mu[f(Y)]$.
EDIT: was thinking of using some kind of generalized Stein's lemma, but this would only work for $f \in \mathcal C_b^1(\mathbb R)$.