  We extend the work of Aarts et al., including an imaginary chemical potential
for quark number into the Sakai-Sugimoto model and codimension k defect
theories. The phase diagram of these models are a function of three parameters,
the temperature, chemical potential and the asymptotic separation of the
flavour branes, related to a mass for the quarks in the boundary theories. We
compute the phase diagrams and the pressure due to the flavours of the theories
as a function of these parameters and show that there are Roberge-Weiss
transitions in the high temperature phases, chiral symmetry restored for the
Sakai-Sugimoto model and deconfined for the defect models, while at low
temperatures there are no Roberge-Weiss transitions. In all the models we
consider the transitions between low and high temperature phases are first
order, hence the points where they meet the Roberge-Weiss lines are triple
points. The pressure for the defect theories scales in the way we expect from
dimensional analysis while the Sakai-Sugimoto model exhibits unusual scaling.
We show that the models we consider are analytic in \mu^2 when \mu^2 is small.
