  The phenomenology of primordial black hole (PBH) physics and the associated
PBH abundance constraints, can be used in order to probe the physics of the
early Universe. In this work, we investigate the PBH formation during the
standard radiation-dominated era by studying the effect of an early F(R)
modified gravity phase with a bouncing behavior which is introduced to avoid
the initial spacetime singularity problem. In particular, we calculate the
energy density power spectrum at horizon crossing time and then we extract the
PBH abundance in the context of peak theory as a function of the parameter
$\alpha$ of our $F(R)$ gravity bouncing model at hand. Interestingly, we find
that in order to avoid GW overproduction from an early PBH dominated era before
Big Bang Nucleosynthesis (BBN), $\alpha$ should lie within the range
$\alpha\leq 10^{-19}M^2_\mathrm{Pl}$. This constraint can be translated to a
constraint on the energy scale at the onset of the Hot Big Bang (HBB) phase,
$H_\mathrm{RD}\sim \sqrt{\alpha}/2$ which can be recast as $H_\mathrm{RD}<
10^{-10}M_\mathrm{Pl}$.
