Exponential corrected thermodynamics of black holes

Recently, it is reported that thermal fluctuations, which are interpreted as quantum effects, modify the black holes entropy by an exponential term. We now find the effect of such modification on the black hole mass and other thermodynamics quantities. We find that the Schwarzschild black hole mass decreased by thermal fluctuations. Hence, we study exponential corrected thermodynamics and statistics of black holes by computing the partition function. We obtain the special condition on the event horizon radius to satisfy Smarr-Gibbs-Duhem relation in the presence of quantum correction. As we know, the Schwarzschild black hole is unstable, while the effect of exponential correction is the stability of $4D$ Schwarzschild black hole as well as the Schwarzschild-AdS black hole at a small area. On the other hand, a $5D$ Schwarzschild black hole is completely unstable. The effect of the quantum correction on the Reissner-Nordstr\"{o}m black hole is instability at quantum scales. Finally, we consider the most general case of charged AdS black hole and study the corrected thermodynamics.