Higher Order Equation of Motions from Lagrangian / Action

I’d like to derive the equations of motion for a scalar field in a FLRW universe, where the metric, as well as the field, are perturbed. I think I should get scalar as well as tensorial expressions.

An abstract expression would suffice, e.g. for the background:

$$
frac{partial L}{partialphi}-frac{mathrm{d}}{mathrm{d} t}frac{partial L}{partialdot{phi}}=0
$$

The scalar field perturbation would depend on the full spacetime, whereas the background field depends only on time.

I think I should get something like

$$
frac{partial^2 L}{partialphi^2}deltaphi-frac{mathrm{d}}{mathrm{d} t}left(frac{partial^2 L}{partialdot{phi}^2}dot{deltaphi}right)=0,
$$

but I am not sure, how the spatial derivatives of the scalar field perturbation come into play here and if the second term is correct.

How do I derive the higher order equations of motion from a Lagragian?