  We study the holographic realization of optimized correlation measures --
measures of quantum correlation that generalize elementary entropic formulas --
in two-dimensional thermal states dual to spacetimes with a black hole horizon.
We consider the symmetric bipartite optimized correlation measures: the
entanglement of purification, Q-correlation, R-correlation, and squashed
entanglement, as well as the mutual information, a non-optimized correlation
measure, and identify the bulk surface configurations realizing their geometric
duals over the parameter space of boundary region sizes and the black hole
radius. This parameter space is divided into phases associated with given
topologies for these bulk surface configurations, and first-order phase
transitions occur as a new topology of bulk surfaces becomes preferred. The
distinct phases can be associated with different degrees of correlation between
the boundary regions and the thermal environment. The Q-correlation has the
richest behavior, with a structure of nested optimizations leading to two
topologically distinct bulk surface configurations being equally valid as
geometric duals at generic points in the phase diagram.
