Density of $C_c^\infty((0,T)\times R^d)$ in $L^2(0,T;H^1)$

Question

I have been dealing with the following problem I can't find an answer to: Let $H^1$ be the usual Sobolev space and $L^2(0,T;H^1)$ the Bochner space of square integrable functions $[0,T]\to H^1$. I know that $C_c^\infty(R^d)$ is dense in $H^1$. But is it also true that $C_c^\infty((0,T)\times R^d)$ is dense in $L^2(0,T;H^1)$? Any hints or references are appreciated.