Start with Einstein's equation
\begin{equation}
R_{ab}-\frac{1}{2} g_{ab} R = \kappa\,T_{ab}.
\tag{1}
\end{equation}
Contract both sides of (1) with $g^{ab}$ to get
$$
R-\frac{N}{2} R = \kappa\,T
\tag{2}
$$
where $N$ is the number of spacetime dimensions and $T$ denotes the trace of $T_{ab}$, just like $R$ denotes the trace of $R_{ab}$. Use (2) in (1) to get
$$
R_{ab} = \kappa\,\left(T_{ab}+ \frac{g_{ab}}{2-{N}}T\right).
\tag{3}
$$
In the weak-field approximation and with a convenient choice of gauge (as described in many textbooks), we have
$$
R_{ab}\approx -\frac{1}{2}\partial^2 h_{ab}
\tag{4}
$$
with $h_{ab}\equiv g_{ab}-\eta_{ab}$. Using this in (3) gives
$$
-\frac{1}{2}\partial^2 h_{ab}\approx \kappa\,\left(T_{ab}+ \frac{g_{ab}}{2-{N}}T\right).
\tag{5}
$$
In the usual approximation with $T_{00}$ being the only significant component of $T_{ab}$, we have
$$
T\approx T_{00}.
\tag{6}
$$
Use this in (5) to get
$$
-\frac{1}{2}\partial^2 h_{ab}\approx \kappa\,\left(T_{ab}+ \frac{g_{ab}}{2-{N}}T_{00}\right).
\tag{7}
$$
In particular, equation (7) implies
$$
-\frac{1}{2}\partial^2 h_{00}\approx \kappa\,\frac{3-{N}}{2-{N}}T_{00}
\tag{8}
$$
and
$$
-\frac{1}{2}\partial^2 h_{jk}\approx -\kappa\,\frac{\delta_{jk}}{2-{N}}T_{00}
\tag{9}
$$
for $j,k\neq 0$. For the physically-relevant case $N=4$, equations (8)-(9) are consistent with
$$
h_{jk}\approx \delta_{jk}h_{00}.
\tag{10}
$$
This is consistent with the equation written in the OP, with $2\phi\equiv h_{00}$.
