Deriving vacuum FRW equations directly from action

[jcap]

Using the Einstein-Hilbert action for a Universe with just the cosmological constant $\Lambda$:
$$S=\int\Big[\frac{R}{2}-\Lambda\Big]\sqrt{-g}\ d^4x$$
I would like to derive the equations of motion:
$$\Big(\frac{\dot a}{a}\Big)^2+\frac{k}{a^2}=\frac{\Lambda}{3}\tag{1}$$
$$2\frac{\ddot a}{a}+\Big(\frac{\dot a}{a}\Big)^2+\frac{k}{a^2}=\Lambda\tag{2}$$
I use the FRW metric to substitute in
$$R=\frac{6}{a^2}(a\ddot a+\dot a^2+k)$$
and
$$\sqrt{-g} \propto a^3$$
I then have the following Euler-Lagrange equation for derivatives of $a(t)$:
$$\frac{\partial L}{\partial a}-\frac{d}{dt}\frac{\partial L}{\partial \dot a}+\frac{d^2}{dt^2}\frac{\partial L}{\partial \ddot a}=0$$
This gives me equation (2).

How would I get equation (1) using this approach?

 

[PeterDonis]

I use the FRW metric to substitute in

This is backwards from the usual approach. The usual approach is to derive the Einstein Field Equation in general form as the Euler-Lagrange equation for the action you gave; this can be done without making any assumption at all regarding the metric.

Once you have the Einstein Field Equation, you then just solve it with appropriate assumptions for the symmetries of the spacetime and a corresponding choice of coordinates.

Note that, for a universe with just a cosmological constant, the general solution is de Sitter spacetime, for which different coordinate choices will give you different values of $k$.