# Is the Palatini-Lovelock action of order $k$ topological in $2k$ dimensions?

I am interested in Lovelock actions in the metric-affine (or Palatini) formalism. It is well-known that the metric version (starting from the Levi-Civita curvature) of the Lovelock lagrangian of order $$k$$ is a topological term in $$2k$$ dimensions. For example, the case $$k=2$$ is the Gauss-Bonnet term.

I have also read in texts on Poincaré gauge gravity (now the connection contains torsion but not non-metricity) that this is also true for torsionful-Lovelock actions.

But some people extrapolate this to the general metric-affine-Lovelock action (with torsion and non-metricity, i.e. a completely general affine connection), but I cannot find the proof anywhere.

Is this true in general? Is it easy to prove?

If it were true:

What does it mean to be "topological" in this particular context? [See Two definitions of topological terms in field theory ] The metric theory can be written as a total derivative; but I think there is no way to do so in the metric-affine case (expanding the connection 2-form using the Cartan's structure equation and trying to extract one of the exterior derivatives), so it must be topological in the other sense. I am confused.

The idea behind the proof is that for boundary terms Lagrangians, the equations of motion are identically satisfied for any configuration of the fields (they are indeed trivial identities, $$0=0$$). And, for metric-affine (or Palatini) Lovelock theories in their critical dimension (when $$D=2k$$), it is always possible to find configurations of the fields that violate at least one equation of motion. Here we provide the counterexamples: https://arxiv.org/abs/1907.12100