Solution
= Solution
Put $K=8\pi G/(3c^2)$. Differentiating $Q=K\rho a^2-\dot a^2$ gives
$$
\dot Q
=K(a^2\dot\rho+2a\dot a\rho)-2\dot a\ddot a.
$$
The continuity equation and the <Friedmann acceleration equation> imply
$$
a^2\dot\rho+2a\dot a\rho
=-a\dot a(\rho+3P)
$$
and
$$
-2\dot a\ddot a
=K a\dot a(\rho+3P).
$$
These terms cancel, so
$$
\boxed{\dot Q=0}.
$$
This is the <First integral of the Friedmann acceleration equation>.