Einstein’s equations relate the geometry of the space-time, encoded in the Einstein tensor to the content of the space-time, encoded in the stress-energy tensor. More precisely, the Einstein’s equation state that the Einstein tensor is equal to the stress tensor (up to a coupling constant). In other words, it is a linear relation between the stress tensor and the geometric tensor. One may wonder why should the relation be linear… After all, Einstein’s equations are non-linear in the metric… Let us first remind how this relation was derived.
The basic idea was to relate the space-time geometry to the matter/energy content. It is clear from construction that the stress tensor describes the second object, i.e. the matter/energy. The (total) stress tensor obeys conservations laws, i.e. it is free of divergence. The question was then, what is the object that can be built out of the metric and at most 2 of its derivatives that is also divergence free ? The answer is: a linear combination of the Einstein tensor and the metric.
It is then very tempting to identify the two objects by a linear relation, with the proportionality coefficient fixed by the small mass (Newtonian) limit. This is what Einstein did, and the big lines of how he derived the celebrated equations named after him. The question is: why a linear coupling ? In fact non-linear coupling has many interesting effects, see my paper here and here.
It can be shown that it is possible to build a divergence free tensor involving the stress tensor (and derivatives of the stress-tensor) that is still divergence-free. This was shown explicitly by one of my collaborators in this paper, up to second order. At this order, it is found that the theory built this way, dubbed gravity with auxiliary fields, is equivalent to a mixture between the so-called Eddington-inspired Born Infeld (EiBI) model or the better known Palatini f(R) model.
An interesting property of models with these non-linear coupling is that they have some singularity avoiding properties (see the Banados-Ferreira paper and my papers with Cardoso and Pani). They further have some deeper relations with bi-metric theories and canonical quantum gravity, see this paper.