  Exploring the significant impacts of topological charge on the holographic
phase transitions and conductivity we start from an Einstein - Maxwell system
coupled with a charged scalar field in Anti - de Sitter spacetime. In our set
up, the corresponding black hole (BH) is chosen to be the topological AdS one
where the pressure is identified with the cosmological constant. Our numerical
computation shows that the process of condensation is favored at finite
topological charge and, in particular, the pressure variation in the bulk
generates a mechanism for changing the order of phase transitions in the
boundary: the second order phase transitions occur at pressures higher than the
critical pressure of the phase transition from small to large BHs while they
become first order at lower pressures. This property is confirmed with the aid
of holographic free energy. Finally, the frequency dependent conductivity
exhibits a gap when the phase transition is second order and when the phase
transition becomes first order this gap is either reduced or totally lost.
