Herein, the spin-forbidden cooling of a gallium hydride molecule is investigated using ab initio quantum chemistry. The cooling transition and the corresponding potential energy curves including , a³Π_0⁻, a³Π_0⁺, a³Π₁, a³Π₂, A¹Π₁, , 1³Σ⁺₁, , , and 2³Σ⁺₁ states are simulated based on the multi-reference configuration interaction approach plus Davidson corrections method. By solving the nuclear Schrödinger equation, we calculate the spectroscopic constants of these states, which are in good agreement with the available experimental values. Based on the transition data, there seems to be a theoretical puzzle: highly diagonally distributed Franck–Condon factor f₀₀ for transitions , , and for the gallium hydride molecule but the intervening state A¹Π₁ for transition is prohibitive to laser cooling. In addition, the transition does not have a suitable rate of optical cycling owing to a large radiative lifetime for state. Our theoretical simulation indicates the solution to the puzzle: the transition has a high emission rate, and there is a suitable radiative lifetime for a³Π₁ state, which can ensure rapid and efficient laser cooling of gallium hydride. The proposed laser drives transition by using three wavelengths (main pump laser λ₀₀; two repumping lasers λ₁₀ and λ₂₁). These results demonstrate the possibility of laser-cooling the gallium hydride molecule, and a sub-microkelvin cool temperature can be reached for this molecule.