homework and exercises - Constraints on $p$ and $rho$ from cosmological metric and $T_{; nu}^{mu nu} = 0$

I'm given the cosmological metric

$$ds^2 = -dt^2 + a(t)^2dx^2 +a(t)^2dy^2 + a(t)^2dz^2$$

and perfect fluid with stress-energy tensor

$$T^{\alpha \beta}=\left(\begin{array}{cccc}{\rho} & {0} & {0} & {0} \\ {0} & {a^{-2} p} & {0} & {0} \\ {0} & {0} & {a^{-2} p} & {0} \\ {0} & {0} & {0} & {a^{-2} p}\end{array}\right)$$

I want to examine $T_{; \nu}^{\mu \nu} = 0$ to determine the conditions imposed on $p$ and $\rho$.

My thought process is this: evaluate the time and spatial components separately. I have also calculated the non-vanishing Christoffel symbols from the given metric: $\Gamma_{x x}^{t}=a(t) \dot{a}(t)$, $\Gamma_{y y}^{t}=a(t) \dot{a}(t)$, $\Gamma_{z z}^{t}=a(t) \dot{a}(t)$, $\Gamma_{t x}^{x}=\dot{a}(t)/a(t)$, $\Gamma_{t y}^{y}=\dot{a}(t)/a(t)$, $\Gamma_{t z}^{z}=\dot{a}(t)/a(t)$. I know this is also related to the Friedmann equations.

I am just unsure of how to calculate $T_{; \nu}^{\mu \nu} = 0$ and then find the constraints on $p$ and $\rho$. Thanks!