The feasibility of spin-forbidden cooling of the InH molecule is investigated based on ab initio quantum chemistry calculations. The potential energy curves for the X1Σ0++, a3Π0-, a3Π0+, a3Π1, a3Π2, A1Π1, 13Σ0-+, and 13Σ1+ states of InH are obtained based on multi-reference configuration interaction plus the Davidson corrections method. The calculated spectroscopic constants are in good agreement with the available experimental data. In addition, the influences of the active space and spin-orbit coupling effects on the potential energy curves and spectroscopic constants are also studied. For Re of a3Π0-, a3Π0+, a3Π1, and a3Π2 states, the error from large active space is small. The potential energy curve of the A1Π1 state is not smooth for small active space. The spin-orbit coupling effects have great influences on the potential well depth and equilibrium internuclear distance of the A1Π state. The Franck-Condon factors and radiative lifetimes are obtained on the basis of the transition dipole moments of the a3Π0+→X1Σ0++,a3Π1→X1Σ0++,and A1Π1→X1Σ0++ transitions. Our calculation indicates that the a3Π1(v'=0)→X1Σ0++(v=0) transition provides a highly diagonally distributed Franck– Condon factor and a short radiative lifetime for the a3Π1 state, which can ensure rapid and efficient laser cooling of InH. The proposed laser drives a3Π0+→X1Σ0++ transitions by using three wavelengths.