  $N$-jettiness subtractions provide a general approach for performing
fully-differential next-to-next-to-leading order (NNLO) calculations. Since
they are based on the physical resolution variable $N$-jettiness,
$\mathcal{T}_N$, subleading power corrections in $\tau=\mathcal{T}_N/Q$, with
$Q$ a hard interaction scale, can also be systematically computed. We study the
structure of power corrections for $0$-jettiness, $\mathcal{T}_0$, for the
$gg\to H$ process. Using the soft-collinear effective theory we analytically
compute the leading power corrections $\alpha_s \tau \ln\tau$ and $\alpha_s^2
\tau \ln^3\tau$ (finding partial agreement with a previous result in the
literature), and perform a detailed numerical study of the power corrections in
the $gg$, $gq$, and $q\bar q$ channels. This includes a numerical extraction of
the $\alpha_s\tau$ and $\alpha_s^2 \tau \ln^2\tau$ corrections, and a study of
the dependence on the $\mathcal{T}_0$ definition. Including such power
suppressed logarithms significantly reduces the size of missing power
corrections, and hence improves the numerical efficiency of the subtraction
method. Having a more detailed understanding of the power corrections for both
$q\bar q$ and $gg$ initiated processes also provides insight into their
universality, and hence their behavior in more complicated processes where they
have not yet been analytically calculated.
