Modified gravity one-loop partition function

The one-loop partition function of the $f(R,R_{\mu\nu}R^{\mu\nu})$ gravity theory is obtained around AdS$_4$ background. After suitable choice of the gauge condition and computation of the ghost determinant, we obtain the one-loop partition function of the theory. The traced heat kernel over the thermal quotient of AdS$_4$ space is also computed and the thermal partition function is obtained for this theory. We have then consider quantum corrections to the thermodynamical quantities in some special cases.