The cubic Palatini gravity action is BRST-consistent at one loop around flat space only after finite counterterms beyond minimal subtraction are added.
One-loop divergences in first order Einstein-Hilbert gravity
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One-loop counterterms are computed in the first order formalism for the Einstein-Hilbert action with a minimally coupled scalar field using the background field method and the heat kernel technique. The {\em off-shell} divergent piece in the harmonic gauge is {\em exactly} the same as the one first found by 't Hooft and Veltman.
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One loop analysis of the cubic action for gravity
The cubic Palatini gravity action is BRST-consistent at one loop around flat space only after finite counterterms beyond minimal subtraction are added.