When entering the heliosphere, Galactic cosmic rays (GCRs) are influenced by magnetic turbulence and Solar wind disturbances, which cause the so-called "solar modulation" effect. Understanding the time-dependent relationship between the Sun's variability and GCR flux modulation is essential for the investigation of the GCR transport processes in the heliosphere, as well as for the establishment of predictive models of GCR radiation in the interplanetary space. The known anti-correlation between GCR flux and sunspot number appears to be delayed by several months, but the origin of such a time lag is unclear. In this work, we are perform the first global characterization of the time lag evolution over the solar cycles and its energy dependence. We made use of a large collection of time-resolved data, both from space missions and ground based observatories. Since the long-term variation of the GCR flux originates by a combination of several physics processes, the investigation presented here may reveal important aspects of the GCR transport in the heliospheric plasma.