Eigen solutions of the SchrÓ§dinger equation and the thermodynamic stability of the black hole temperature

The approximate analytical solutions of the SchrÓ§dinger equation for Eckart potential is obtained via supersymmetry shape invariance approach. The energy equation and the corresponding wave function are obtained in a closed and compact form. The wave function was used to calculate the Rényi entropy. The results of the Rényi entropy was used to study the mass energy parameter, temperature and heat capacity of the black hole. From the results obtained, the temperature of the black hole becomes stable as the two Eckart potential parameters increases respectively.