  Higher curvature Lovelock gravity theories can have a number of maximally
symmetric vacua with different values of the curvature. Critical surfaces in
the space of Lovelock couplings separate regions with different numbers of such
vacua, and there exist symmetry breaking regions with no maximally symmetric
vacua. Especially in such regimes, it is interesting to ask what reduced
symmetry vacua may exist. We study this question, focusing on vacua that are
products of maximally symmetric spaces. For low order Lovelock theories, we
assemble a map of such vacua over the Lovelock coupling space, displaying
different possibilities for vacuum symmetry breaking. We see indications of
interesting structure, with e.g. product vacua in Gauss-Bonnet gravity covering
the entirety of the symmetry breaking regime in $5$-dimensions, but only a
limited portion of it in $6$-dimensions.
