The most commonly used tools are Lagrangians and Hamiltonians and are used to derive the Einstein field equations.
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The classic solution of the Einstein field equations that describes a homogeneous and isotropic universe is called the Friedmann–Lemaître–Robertson–Walker metric, or FLRW, after Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker, who worked on the problem in 1920's and 30's independently of Friedmann.
If one is going to solve the Einstein field equations using approximate methods such as the Post-Newtonian expansion, then one should try to choose a coordinate condition which will make the expansion converge as quickly as possible (or at least prevent it from diverging).