Chao-Ming Zhang, Yun Soo Myung, Lina Zhang, De-Cheng Zou, Fu-Wen Shu

We investigate how a negative cosmological constant affects the Gauss-Bonnet (GB) scalarization in the Einstein-Maxwell-scalar-Gauss-Bonnet theory with a scalar coupling constant $η$ to GB term. We focus on the instability of Reissner-Nordström-AdS (RN-AdS) black holes under a scalar perturbation governed by an effective mass $μ^2_{\text{eff}}$ sourced by the GB term. Unlike the asymptotically flat spacetime case, the onset of scalarization is not merely determined by $μ^2_{\text{eff}} < 0$, but it is constrained by the Breitenlohner-Freedman (BF) bound. In case that the BF bound is violated ($η>2.25$ with $Λ=-0.5$), one may find AdS-tachyoinic instability. We find that for $0<η<2.25$, the GB$^+$ scalarization may be performed through spontaneous scalarization, while for $η<0$ the GB$^-$ scalarization is found to give the single branch of scalarized AdS black holes. For the GB$^+$ scalarization in $η_{th}\leη<2.25$ with $η_{th}$ threshold instability, we obtain the single branch ($n=0$ fundamental branch) of scalarized AdS black holes, contradicting to infinite branches. Finally, it is observed from Gibbs free energy that a phase transition from scalarized AdS black holes to RNAdS black hole occur natually and it is confirmed to be second-order.