We complete the formulation of the standard model of first star formation by exploring the possible impact of $mathrm{LiH}$ cooling, which has been neglected in previous simulations of non-linear collapse. Specifically, we find that at redshift $zgtrsim 5$, the cooling by $mathrm{LiH}$ has no effect on the thermal evolution of shocked primordial gas, and of collapsing primordial gas into minihaloes or relic HII regions, even if the primordial lithium abundance were enhanced by one order of magnitude. Adding the most important lithium species to a minimum network of primordial chemistry, we demonstrate that insufficient $mathrm{LiH}$ is produced in all cases considered, about $[mathrm{LiH/Li}]sim 10^{-9}$ for $Tlesssim 100$ K. Indeed, $mathrm{LiH}$ cooling would only be marginally significant in shocked primordial gas for the highly unlikely case that the $mathrm{LiH}$ abundance were increased by nine orders of magnitude, implying that $all$ lithium would have to be converted into $mathrm{LiH}$. In this study, photo-destruction processes are not considered, and the collisional disassociation rate of $mathrm{LiH}$ is possibly underestimated, rendering our results an extreme upper limit. Therefore, the cooling by $mathrm{LiH}$ can safely be neglected for the thermal evolution of Population~III star-forming gas.
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