  We explore the phase structure for defect theories in full generality using
the gauge/gravity correspondence. On the gravity side, the systems are
constructed by introducing M (probe) D(p+4-2k)-branes in a background generated
by N Dp-branes to obtain a codimension-k intersection. The dual gauge theory is
a U(N) Supersymmetric Yang-Mills theory on a (1+p-k)-dimensional defect with
both adjoint and fundamental degrees of freedom. We focus on the phase
structure in the chemical potential versus temperature plane. We observe the
existence of two universality classes for holographic gauge theories, which are
identified by the order of the phase transition in the interior of the chemical
potential/temperature plane. Specifically, all the sensible systems with no
defect show a third order phase transition. Gauge theories on a defect with
(p-1)-spatial directions are instead characterised by a second order phase
transition. One can therefore state that the order of this phase transition is
intimately related to the codimensionality of the defect. We also discuss the
massless hypermultiplet at low temperature, where a thermodynamical instability
seems to appear for $p<3$. Finally, we comment on such an instability.
