Quantum State Dependence of Thermodynamic Phase Transition in 4D AdS Gauss-Bonnet Quantum Black Holes Surrounded With Cloud of Strings

According to the Lovelock theorem where the model \cite{Glav} could not applicable for Einstein Gauss Bonnet (EGB) gravity in all 4D curved spacetimes, authors of the reference \cite{ali} presented an effective model by applying break of diffeomorphism property. Hence we use the latter model instead of former for study of thermodynamic behavior of a 4D AdS EGB spherically symmetric static black hole which surrounded with a cloud of string. In short our work is extension of the works given by \cite{Veer,Heg} but not by using \cite{Glav} but by applying \cite{ali}. Our metric solutions are obtained versus the Hermite polynomials (quantum harmonic Oscillator) for which eigen values come from single scale defined by multiplication of the coupling constants of the model: Namely the regularized GB parameter, AdS radius, the black hole ADM mass and the string tension. Hence we claim the obtained metric solution is in fact behavior of quantized black hole. Because the GB term is originated from renormalization of quantum matter fields. Also we should pointed that this kind of quantization is different with the canonical quantization (Wheeler De Witt). Our study shows that all phase transitions of this quantum black hole are dependent to the Hermite quantum numbers.