NANOGrav 15-year data hint towards an alternative inflationary scenario, as opposed to the usual slow-roll one (as the usual slow-roll inflation predicts a low amplitude of gravitational waves around the NANOgrav frequency range), for describing the early time dynamics of the universe. The modified inflation should be such that it leads to a blue-tilted primordial tensor power spectrum around the NANOGrav frequency range, but maintaining the Planck constraints over the CMB scales. Regarding the constraint over CMB scales, the recent the Atacama Cosmology Telescope (combined with the Planck 2018 and BAO) refines the constraint on inflationary observables, compared to the Planck-only 2018 measurements. Based on these observations, we are interested to simultaneously fit these two through the thermodynamic route of cosmology. The entropy of the apparent horizon is considered to be four-parameter generalized entropy (which is able to generalize the other known entropies proposed so far). In particular, we will show that four-parameter generalized entropy proves to be compatible with these observations for common range of entropic parameters. After the inflation ends, the entropic energy density (coming from the generalized entropy) decays to relativistic particles and leads to standard Big Bang cosmology. Therefore we argue that if the future observatories can detect the signal of primordial GWs, then our theoretical expectation carried in the present work may provide a possible testbed for generalized entropic cosmology.