  Quantum fluctuation consequences have significant role in high-energy
physics. These fluctuation often regarded as a correction of the infrared (IR)
limit. Such correction contribute to the high-energy limit of thermodynamical
quantities and the stability conditions of black holes. In this work, we
analyze the thermal stability of black holes in the presence of thermal
fluctuations. We consider AdS black hole in Born-Infeld massive gravity with
non-abelian hair and the charged AdS black hole with a global monopole. We
develop many thermodynamical quantities such as entropy, temperature, pressure,
heat capacity of a system at constant volume and pressure, ratio between the
heat capacities at constant pressure and volume, Gibbs free energy and
Helmholtz free energy for both black holes. The critical behavior and phase
transitions of black holes are also presented. We also observe the local and
global stability of black holes in the grand canonical ensemble and canonical
ensemble for the specific values of different parameters, such as, symmetry
breaking parameter $\eta$, massive parameter $m$ and non-abelian hair $\nu$.
