  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.
