  Matched asymptotic expansion is a useful technique in General Relativity and
other fields whenever interaction takes place between physics at two different
length scales. Here matched asymptotic expansion is argued to be equivalent
quite generally to Classical Effective Field Theory (CLEFT) where one (or more)
of the zones is replaced by an effective theory whose terms are organized in
order of increasing irrelevancy, as demonstrated by Goldberger and Rothstein in
a certain gravitational context. The CLEFT perspective has advantages as the
procedure is clearer, it allows a representation via Feynman diagrams, and
divergences can be regularized and renormalized in standard field theoretic
methods. As a side product we obtain a wide class of classical examples of
regularization and renormalization, concepts which are usually associated with
Quantum Field Theories.
  We demonstrate these ideas through the thermodynamics of caged black holes,
both simplifying the non-rotating case, and computing the rotating case. In
particular we are able to replace the computation of six two-loop diagrams by a
single factorizable two-loop diagram, as well as compute certain new three-loop
diagrams. The results generalize to arbitrary compactification manifolds. For
caged rotating black holes we obtain the leading correction for all
thermodynamic quantities. The angular momentum is found to non-renormalize at
leading order.
