  A new class of N=2 locally supersymmetric higher-derivative invariants is
constructed based on logarithms of conformal primary chiral superfields. They
characteristically involve a coupling to R_{\mu\nu}^2 - 1/3*R^2, which equals
the non-conformal part of the Gauss-Bonnet term. Upon combining one such
invariant with the known supersymmetric version of the square of the Weyl
tensor, one obtains the supersymmetric extension of the Gauss-Bonnet term. The
construction is carried out in the context of both conformal superspace and the
superconformal multiplet calculus. The new class of supersymmetric invariants
resolves two open questions. The first concerns the proper identification of
the 4D supersymmetric invariants that arise from dimensional reduction of the
5D mixed gauge-gravitational Chern-Simons term. The second is why the pure
Gauss-Bonnet term without supersymmetric completion has reproduced the correct
result in calculations of the BPS black hole entropy in certain models.
