  We study the extended thermodynamics, obtained by considering the
cosmological constant as a thermodynamic variable, of STU black holes in
4-dimensions in the fixed charge ensemble. The associated phase structure is
conjectured to be dual to an RG-flow on the space of field theories. We find
that for some charge configurations the phase structure resembles that of a Van
der Waals gas: the system exhibits a family of first order phase transitions
ending in a second order phase transition at a critical temperature. We
calculate the holographic entanglement entropy for several charge
configurations and show that for the cases where the gravity background
exhibits Van der Waals behavior, the entanglement entropy presents a transition
at the same critical temperature. To further characterize the phase transition
we calculate appropriate critical exponents and show that they coincide. Thus,
the entanglement entropy successfully captures the information of the extended
phase structure. Finally, we discuss the physical interpretation of the
extended space in terms of the boundary QFT and construct various holographic
heat engines dual to STU black holes.
