  We sum the leading logarithms $\alpha_s^n \ln^{2 n-1}(1-z)$, n=1,2,... near
the kinematic threshold $z=m_H^2/\hat{s}\to 1$ at next-to-leading power in the
expansion in (1-z) for Higgs production in gluon fusion. We highlight the new
contributions compared to Drell-Yan production in quark-antiquark annihilation
and show that the final result can be obtained to all orders by the
substitution of the colour factor $C_F\to C_A$, confirming previous fixed-order
results and conjectures. We also provide a numerical analysis of the
next-to-leading power leading logarithms, which indicates that they are
numerically relevant.
