Stellar and AGN feedback primarily affect the formation and evolution of galaxies and the circumgalactic medium, leaving some imprint on larger scales. Based on the SIMBA hydrodynamical simulation suite, and using the full set of Minkowski functionals (MFs), this study systematically analyses the time evolution of the global geometry and topology of the gas temperature, pressure, density (total, HI and H$_2$), and metallicity fields between redshift $z=5$ and $z=0$. MFs show that small-scale astrophysical processes are persistent and manifest on larger, up to tens of Mpc scales, highlighting the specific morphological signatures of the relevant feedback mechanisms on such scales in the last $\sim12$~Gyr. In qualitative terms, one can establish a ranking that varies according to the field considered: stellar feedback mostly determines the morphology of the pressure and density fields, AGN jets are the first cause for the morphology of the temperature and metallicity fields, while X-ray heating and AGN winds play the second most important role in shaping the geometry and topology of all the gaseous fields but metallicity. Hence the cosmic evolution of the geometry and topology of fields characterising the thermodynamical and chemical properties of the cosmic web provides complementary larger-scale constraints to galaxy formation models.