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We explore the phase structure for defect theories in full generality usingthe gauge/gravity correspondence. On the gravity side, the systems areconstructed by introducing M (probe) D(p+4-2k)-branes in a background generatedby N Dp-branes to obtain a codimension-k intersection. The dual gauge theory isa U(N) Supersymmetric Yang-Mills theory on a (1+p-k)-dimensional defect withboth adjoint and fundamental degrees of freedom. We focus on the phasestructure in the chemical potential versus temperature plane. We observe theexistence of two universality classes for holographic gauge theories, which areidentified by the order of the phase transition in the interior of the chemicalpotential/temperature plane. Specifically, all the sensible systems with nodefect show a third order phase transition. Gauge theories on a defect with(p-1)-spatial directions are instead characterised by a second order phasetransition. One can therefore state that the order of this phase transition isintimately related to the codimensionality of the defect. We also discuss themassless hypermultiplet at low temperature, where a thermodynamical instabilityseems to appear for $p<3$. Finally, we comment on such an instability.

Universality of Holographic Phase Transitions and Holographic Quantum Liquids