Variance of the exit time from the band of width 4

A standard Brownian motion $B$ starts at $0$. Let $\tau$ be the first time $|B|$ exceeds $4$. Find the variance of $\tau$.

$512/3$. For exit level $a$, optional stopping on $B_t^2-t$ gives $\mathbb{E}[T_a]=a^2$, and on the martingale $B_t^4-6tB_t^2+3t^2$ gives $\mathbb{E}[T_a^2]=\tfrac53a^4$. So $\operatorname{Var}(T_a)=\tfrac23a^4=\tfrac23\cdot256$.