Is there a way to conceive a TOV equation, and therefore the stability analysis for a metric like:
$$ ds^2 = -dt^2 + a^2(t,r)\big(dr^2 + r^2d\Omega ^2\big)~?\tag{1}$$
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The TOV equations apply when you have a static, spherically symmetric spacetime (they describe the hydrostatic equilibrium): they are derived from the Einstein equations under the assumption of static spherical spacetime and perfect fluid, see also this.
If the body is collapsing /exploding /oscillating in a spherically symmetric way, you need to consider the full Einstein equations (they simplify a lot because of spherical symmetry) and provide a model for matter (the simplest possibility is the perfect fluid, like for the TOV). You can find a scheme in Simple equations for general relativistic hydrodynamics in spherical symmetry applied to neutron star collapse. Another scheme is, e.g., the one implemented in the one-dimensional, general-relativistic hydrodynamic code (publicly available) developed in this paper. A simpler case was studied by Snyder and Oppenheimer: calculated the gravitational collapse of a pressure-free sphere of dust particles as described by general relativity.
Edit: I notice now that maybe you are more interested in a FLRW spacetime, while the links in my answer are for a Schwarzschild-like metric (like the one of a non-rotating neutron star/white dwarf). I am a bit confused on what exactly you are looking for. However, the general comments are still valid: you just have to provide a matter model (I.e. an energy-momentum tensor) and write explicitly Einstein equations for you metric ansatz.
