  We have investigated the black hole thermodynamics and the phase transition
for renormalized group improved asymptotically safe Schwarzschild black hole.
This geometry takes into account the quantum gravitational correction in the
running gravitational constant identifying $G(r) \equiv G(k=k(r))$. We studied
various thermodynamic quantity like the Hawking temperature, specific heat and
entropy for the general parameter $\gamma$ for quantum corrected Schwarzschild
metric. We have noticed that the coefficient of the leading quantum correction,
that is, the logarithmic correction gets affected by the presence of $\gamma$.
We further study the local temperature, thermal stability of the black hole and
the free energy considering a cavity enclosing the black hole. According to the
local specific heat, there exists three black hole states, among them the large
and tiny black hole are thermally stable states. We further investigate the
on-shell free energy and find that no Hawking-Page phase transition occurs here
unlike the ordinary Schwarzschild black hole. The black hole state always
prevails for all temperatures. Also, we have found two critical points,
$T_{c1}$ and $T_{c2}$, corresponding to the phase transition from one black
hole state to another.

General Relativity and Quantum Cosmology;High Energy Physics - Theory