Interpretation of the first order formalism of f(R)-type gravity and the   corresponding second order formalism
  f(R)-type gravity in the first order formalism is interpreted as Einstein gravity with non-minimal coupling arising from the use of unphysical frame. Identification of the corresponding second order higher-curvature gravity in the physical frame is proposed by requiring that the action is the same. 
Y.Ezawa, H.Iwasaki, Y.Ohkuwa, T.Uegaki, N.Yamada and T.Yano