•New charged state of carbon in β-Ga2O3 are found and corresponding formation energies are given.•Thermodynamics of different substitutions including phonon are explored.•The vacancy concentrations are predicted against oxygen partial pressure.•Electronic and optical properties are predicted, and yellow emissions are firstly observed. Calculations using the revised Heyd-Scuseria-Ernzerhof screened hybrid functional (HSE06) reveal the opto-electronic effects of carbon in β-Ga2O3. Stable charged states with carbon dopant are proved by phonon calculations which can result in the deep levels in the band gap. Stronger distortion can happen around oxygen sites substituted by carbon than those at gallium sites leading to interactions changing. Temperature and oxygen pressure can have huge effect on the formation of substitutions and the electronic conductivity. As partial oxygen pressure increase, the substitution concentration increases, and decrease with temperature increasing. Electronic conductivity of substitution of carbon at oxygen sites can get enhanced hugely, and it increases with temperature increasing. Besides electronic properties, carbon substitution can also have great impact on optical properties. Red-infrared PL emissions can be induced with substitutions at gallium sites, and yellow and blue emissions can be achieved by substitutions with oxygen in different charged states. The work can effectively predict the impact of carbon substitution in β-Ga2O3 and be useful for improvement of device designs and optimizations.