  The quantum scenario concerning Hawking radiation, gives us a precious clue
that a black hole has its temperature directly connected to its area gravity
and that its entropy is proportional to the horizon area. These results have
shown that there exist a deep association between thermodynamics and gravity.
The recently introduced Barrow formulation of back holes entropy, influenced by
the spacetime geometry, shows the quantum fluctuations effects through Barrow
exponent, $\Delta$, where $\Delta=0$ represents the usual spacetime and its
maximum value, $\Delta=1$, characterizes a fractal spacetime. The quantum
fluctuations are responsible for such fractality. Loop quantum gravity approach
provided the logarithmic corrections to the entropy. This correction arises
from quantum and thermal equilibrium fluctuations. In this paper we have
analyzed the nonextensive thermodynamical effects of the quantum fluctuations
upon the geometry of a Barrow black hole. We discussed the Tsallis' formulation
of this logarithmically corrected Barrow entropy to construct the equipartition
law. Besides, we obtained a master equation that provides the equipartition law
for any value of the Tsallis $q$-parameter and we analyzed several different
scenarios. After that, the heat capacity were calculated and the thermal
stability analysis was carried out as a function of the main parameters,
namely, one of the so-called pre-factors, $q$ and $\Delta$.
