We use the AdS/CFT correspondence to study the thermodynamics of massive = 2 supersymmetric hypermultiplet flavor fields coupled to = 4 supersymmetric SU(N_c) Yang-Mills theory, formulated on curved four-manifolds, in the limits of large N_c and large 't Hooft coupling. The gravitational duals are probe D-branes in global thermal AdS. These D-branes may undergo a topology-changing transition in the bulk. The D-brane embeddings near the point of the topology change exhibit a scaling symmetry. The associated scaling exponents can be either real- or complex-valued. Which regime applies depends on the dimensionality of a collapsing submanifold in the critical embedding. When the scaling exponents are complex-valued, a first-order transition associated with the flavor fields appears in the dual field theory. Real scaling exponents are expected to be associated with a continuous transition in the dual field theory. For one example with real exponents, the D7-brane, we study the transition in detail. We find two field theory observables that diverge at the critical point, and we compute the associated critical exponents. We also present analytic and numerical evidence that the transition expresses itself in the meson spectrum as a non-analyticity at the critical point. We argue that the transition we study is a true phase transition only when the 't Hooft coupling is strictly infinite.