We have investigated the consequences of external pressure on the magneto-thermal, critical behavior, and magnetocaloric properties of the polycrystalline alloy Gd5Si2Ge2 and Cu-substitute Gd5Si2Ge1.9Cu0.1 alloy. The ordering temperature of the parent compound is found to be strongly dependent on the external pressure dTc/dP ≅+4.91 K/kbar whereas insensitive for the Cu-substituted alloy (dTc/dP≅+0.47 K/kbar). Modified Arrott plot (MAP) and Kouvel-Fisher (KF) methods have been used to extract the critical exponents to study the critical behavior. The critical behavior of both the polycrystalline alloys is in a close match with the three dimensional (3D)-Heisenberg class of universalities. In the parent alloy, the critical field for metamagnetic transition increases with the pressure. The pressure-induced coupling has a remarkable effect on the refrigeration capacity (RC) (355 J/kg K to 581 J/kg K).