Thermodynamic Analysis of Black Holes with Multiple Horizons in Beyond Horndeski Gravity

We perform thermodynamic analysis of primary and secondary hair black holes found in Beyond Horndeski gravity. It turns out that even though the strong energy condition is violated for primary hair black hole, it is a singular black hole. The primary hair black hole is described completely by three parameters of mass M, action parameter λ, and ξ = ηq5 (action parameter η and scalar charge q). We choose three cases of ξ = 243, 5.51, 0.219 to represent primary hair black hole properly. Four horizons with ξ = 243 make a closed loop in the horizon space. The secondary hair black hole depends on M and λ which becomes a magnetic charge in connection with the regular (Bardeen) black hole. We observe that the action parameter λ plays an important role in describing primary and secondary hair black holes. It is shown that the secondary hair black hole may transit into the primary hair black hole when analyzing heat capacity and Gibbs free energy. However, for two double horizons with ξ=5.51 and 0.219 forming closed loops, their thermodynamic descriptions are quite different from four horizons and thus, they could represent a new feature of primary hair black hole. Finally, the shadow radius analysis is performed to distinguish four horizons from two double horizons by comparing them with the EHT observation.