Many sources have the expression
$$\operatorname{Var}\left(\widehat{\beta}_0+\widehat{\beta}_1 x_0\right)=\sigma^2\left(\frac{1}{n}+\frac{\left(x_0-\bar{x}\right)^2}{\sum_{i=1}^n\left(x_i-\bar{x}\right)^2}\right)$$
where $x_0$ is a new observation. This expression is nice in the way that it shows if $x_0$ is very different from $\bar x$, then the uncertainty of this mean response is higher.
However, what we usually see in the multi-linear regression is Var($x_0\hat\beta$) = $x_0(X^TX)^{-1}x_0$, where now $x_0$ is a vector, and $X$ is the design matrix. Looking at this, it seems the larger the magnitude of $x_0$, the larger the uncertainty of the mean response, which does not make sense.
Is there a similar expression for multi-linear regression where one can interpret the uncertainty of the given mean response estimate by how far $x_0$ is away from $\bar x$?
I'm grateful for any resources. If you know please leave a comment.