Where is the wiggle room in current gravity theories?

To understand where the "wiggle room" in general relativity is it is useful to look at one of the main theorems that constrains GR, Lovelock's theorem. This says that if we start from an action that

  • is local
  • depends only the spacetime metric
  • is at most second order in derivatives of the metric and
  • is in 4 spacetime dimensions then the only possible equation of motion for the metric is the Einstein equation (possibly with cosmological constant).

The conditions of this theorem immediately tell us what assumptions we need to let go off to find alternative theories. You can

  • Consider theories with more fields than just the metric, as is done for example in scalar-tensor theories such as Brans-Dicke
  • Consider actions that contain higher derivatives of the metric. For examples in f(R) gravity.
  • Consider non-local actions (I don't know of any good examples that people study.)
  • Consider theories in different number of dimensions than 4.