Let $\frak g$ be a symmetrizable generalized Kac-Moody algebra and $\frak n_-$ the sum of all its negative root spaces. We obtain a formula for the twining characters of the Lie algebra homology modules of $\frak n_-$ with coefficients in the irreducible highest weight $\frak g$-module $L(\Lambda)$ of symmetric, dominant integral highest weight $\Lambda$. This formula gives a new (and convincing) proof of the formula for the twining character of $L(\Lambda)$ above.