DSolve[{3 (D[a[t], t]^2/a[t]^2 + 1/a[t]^2) == 8 Pi/a[t]^4, a[0] == 1}, a[t], t]

{{a[t] -> Sqrt[3 - 2 Sqrt[3 (-3 + 8 [Pi])] t - 3 t^2]/Sqrt[3]}, {a[t] -> Sqrt[3 + 2 Sqrt[3 (-3 + 8 [Pi])] t - 3 t^2]/Sqrt[3]}}

Plot[Sqrt[3 + 2 Sqrt[3 (-3 + 8 [Pi])] t - 3 t^2]/Sqrt[2], {t, 0, 6}]

sol = NDSolve[{3 (D[a[t], t]^2/a[t]^2 + 1/a[t]^2) == 8 Pi/a[t]^4, a[0] == 1}, a, {t, 0, 6}]

Plot[Evaluate[a[t] /. sol[[2]]], {t, 0, 5.3}]

Method -> {"MethodOfLines",
     "DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> 1}}