  We present a procedure to link the deformation parameter $\beta$ of the
generalized uncertainty principle (GUP) to the two free parameters $\om$ and
$\gamma$ of the running Newtonian coupling constant of the Asymptotic Safe
gravity (ASG) program. To this aim, we compute the Hawking temperature of a
black hole in two different ways. The first way involves the use of the GUP in
place of the Heisenberg uncertainty relations, and therefore we get a deformed
Hawking temperature containing the parameter $\beta$. The second way involves
the deformation of the Schwarzschild metric due to the Newtonian coupling
constant running according to the AS gravity prescription. The comparison of
the two techniques yields a relation between $\beta$ and $\om$, $\gamma$. As a
particular case, we discuss also the so called $\xi$-model. The relations
between $\beta$ and $\om$, $\xi$ allow us to transfer upper bounds from one
parameter to the others.
