Asymptotics of d-dimensional Kaluza-Klein black holes: beyond the   Newtonian approximation
  We study the thermodynamics of small black holes in compactified spacetimes of the form R^(d-1)x S^1. This system is analyzed with the aid of an effective field theory (EFT) formalism in which the structure of the black hole is encoded in the coefficients of operators in an effective worldline Lagrangian. In this effective theory, there is a small parameter $\lambda$ that characterizes the corrections to the thermodynamics due to both the non-linear nature of the gravitational action as well as effects arising from the finite size of the black hole. Using the power counting of the EFT we show that the series expansion for the thermodynamic variables contains terms that are analytic in $\lambda$, as well as certain fractional powers that can be attributed to finite size operators. In particular our operator analysis shows that existing analytical results do not probe effects coming from horizon deformation. As an example, we work out the order $\lambda^2$ corrections to the thermodynamics of small black holes for arbitrary d, generalizing the results in the literature. 
Yi-Zen Chu, Walter D. Goldberger, and Ira Z. Rothstein