  Planar AdS black holes with axionic charge have finite DC conductivity due to
momentum relaxation. We obtain a new family of exact asymptotically AdS$_4$
black branes with scalar hair, carrying magnetic and axion charge, and we study
the thermodynamics and dynamic stability of these, as well as of a number of
previously known electric and dyonic solutions with axion charge and scalar
hair. The scalar hair for all solutions satisfy mixed boundary conditions,
which lead to modified holographic Ward identities, conserved charges and free
energy, relative to those following from the more standard Dirichlet boundary
conditions. We show that properly accounting for the scalar boundary conditions
leads to well defined first law and other thermodynamic relations. Finally, we
compute the holographic quantum effective potential for the dual scalar
operator and show that dynamical stability of the hairy black branes is
equivalent to positivity of the energy density.
