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.
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