  We introduce a holomorphic torsion invariant of log-Enriques surfaces of
index two with cyclic quotient singularities of type $\frac{1}{4}(1,1)$. The
moduli space of such log-Enriques surfaces with $k$ singular points is a
modular variety of orthogonal type %of dimension $10-k$ associated with a
unimodular lattice of signature $(2,10-k)$. We prove that the invariant, viewed
as a function on the modular variety, is given by the Petersson norm of an
explicit Borcherds product. We note that this torsion invariant is essentially
the BCOV invariant in the complex dimension $2$. As a consequence, the BCOV
invariant in this case is not a birational invariant, unlike the Calabi-Yau
case.
