Coefficient variance after stacking two copies of data

In OLS with i.i.d. normal errors of variance $\sigma^2$, $\operatorname{Var}(\hat\beta)=\sigma^2(X^\top X)^{-1}$. If every observation is copied once (so the dataset is the original stacked on itself) and the regression is refit to give $\hat\beta'$, what constant $c$ satisfies $\operatorname{Var}(\hat\beta')=c\operatorname{Var}(\hat\beta)$? Enter $-1$ if there is none.

$1/2$. Stacking the design on itself gives $X'^\top X'=2X^\top X$, so with the same error variance the formula yields half the variance. (The copies are not genuinely new information, which is why this formula overstates precision.)