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 of our 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), should lie within the range . This constraint can be translated to a constraint on the energy scale at the onset of the Hot Big Bang (HBB) phase, which can be recast as .