= Solution
The ordinary <Sachs-Wolfe effect> combines intrinsic last-scattering <temperature> with the gravitational redshift of <photons> escaping the last-scattering <gravitational potential>. For adiabatic large-scale perturbations in matter domination, with the potential convention in the question, the combination is $\Theta=\phi/(3c^2)$, up to the observer monopole. It is distinct from the <Integrated Sachs-Wolfe effect> caused by a time-varying potential along the later <line of sight>.
Use the <Fourier transform> convention in which $P_\delta(k)=Ak$ is the <matter power spectrum> and $\langle\delta_{\mathbf k}\delta_{\mathbf k'}^*\rangle=(2\pi)^3\delta_D(\mathbf k-\mathbf k')P_\delta(k)$. The <Poisson equation> becomes $-k^2\phi_{\mathbf k}=4\pi G\bar\rho_0\delta_{\mathbf k}$, giving
$$
P_\phi(k)=\frac{(4\pi G\bar\rho_0)^2A}{k^3},\qquad
\Delta_\phi^2=\frac{(4\pi G\bar\rho_0)^2A}{2\pi^2c^4}.
$$
With the matter-dominated critical-density approximation $\bar\rho_0=3H_0^2/(8\pi G)$,
$$
\boxed{\Delta_\phi^2=\frac{9AH_0^4}{8\pi^2c^4},}
$$
independent of $k$. Thus the <Harrison-Zeldovich spectrum> gives scale-invariant potential fluctuations and the requested $H_0^4$ dependence. Keeping a general present matter fraction would insert $\Omega_m^2$; the answer here follows the stipulated critical matter-dominated normalization. In this <Fourier transform> convention, $A$ has dimensions of length to the fourth power.
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