This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard action is given (in the absence of matter and with cosmological constant $\Lambda=0$) by

$$ \mathcal{S_{EH}}(g_{\mu\nu}) = \int_M R \sqrt{-g}\mbox{ }d^4x$$

where $M$ is a (compact) differentiable 4-manifold, $g_{\mu\nu}$ is a Lorentzian metric on $M$, $R$ is scalar curvature and $\sqrt{-g}\mbox{ }d^4x$ is the standard volume form. Critical points of this action (with respect to variations in $g_{\mu\nu}$) give Lorentzian metrics which are solutions to Einstein's field equations for general relativity.

QUESTION: Does anyone know of work using a similar action, but where the absolute value $|R|$ appears instead of $R$? That is, I'm interested in references to previous work concerning the action $$ \mathcal{S}(g_{\mu\nu}) = \int_M |R| \sqrt{-g}\mbox{ }d^4x.$$

Given the huge amount of interest in quantum gravity, I would assume that someone has examined this. However, I was unsuccessful in my searches. I'm not a physicist, so perhaps I'm missing some bit of terminology that is standard. Any help pointing me in the right direction would be greatly appreciated!