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Gas cooling processes in the interstellar medium (ISM) are key to understanding how star-formation processes occur in galaxies. Far-infrared (FIR) fine-structure emission lines can be used as a tool to understand the gas conditions and trace the different phases of the ISM. We model the most important far-infrared (FIR) emission lines throughout cosmic time back to $z=6$ with cosmological hydrodynamical simulations. We study how different physical parameters, such as the interstellar radiation field (ISRF) and metallicity, impact the ISM phases traced by FIR line luminosities and connect those with the star-formation rate (SFR). We implement a physically motivated multi-phase model of the ISM by post-processing EAGLE cosmological simulation with Cloudy look-up tables. In this model, we assume four phases of the ISM: dense molecular gas, neutral atomic gas, diffuse ionised gas (DIG) and HII regions. Our model shows good agreement with the observed luminosity-SFR relation up to $z=6$ in the FIR emission lines analysed and we also provide linear fits. Our predictions also agree with observations in terms of diagnostic diagrams involving various line ratios. We find that [C II] is the best SFR tracer of the FIR lines even though it traces multiple ISM phases, while [O III] and [N II] can be used to understand the DIG-HII balance in the ionised phase. In addition, line ratios like [C II]/[O III] and [N II/[O I] are useful to trace parameters such as ISRF, metallicity and specific star-formation rate. These results help to interpret observations of FIR line emission from the local Universe to high-$z$ galaxies.