= Solution
For one observation with density $f(x,\theta)$, the <Fisher information matrix> at $\theta_0$ is
$$
I(\theta_0)
=\mathbb E_{\theta_0}\!\left[
\nabla_\theta\log f(X,\theta_0)
\nabla_\theta\log f(X,\theta_0)^T
\right].
$$
Here
$$
\log f(x,\theta)=\text{constant}-\frac12\lVert x-\theta\rVert^2,
\qquad
\nabla_\theta\log f(x,\theta)=x-\theta.
$$
Since $\operatorname{Cov}_{\theta_0}(X)=I_p$, the <Fisher information of a multivariate normal location model> is
$$
\boxed{I(\theta_0)=I_p.}
$$
The information in the full independent sample is $nI_p$ by <Tensorization of Fisher information>.
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