  Cross-correlating 21cm with known cosmic signals will be invaluable proof of
the cosmic origin of the first 21cm detections. As some of the widest fields
available, comprising thousands of sources with reasonably known redshifts,
narrow-band Lyman alpha emitter (LAE) surveys are an obvious choice for such
cross-correlation. Here we revisit the 21cm -- LAE cross-correlation, relaxing
the common assumption of reionization occurring in a pre-heated intergalactic
medium (IGM). Using specifications from the Square Kilometre Array and the
Subary Hyper Supreme-Cam, we present new forecasts of the 21cm -- LAE
cross-correlation function at $z\sim7$. We sample a broad parameter space of
the mean IGM neutral fraction and spin temperature, ($\bar{x}_{\rm{HI}}$,
$\bar{T}_{\rm S}$). The sign of the cross-correlation roughly follows the sign
of the 21cm signal: ionized regions which surround LAEs correspond to relative
hot spots in the 21cm signal when the neutral IGM is colder than the CMB, and
relative cold spots when the neutral IGM is hotter than the CMB. The amplitude
of the cross-correlation function generally increases with increasing
$\bar{x}_{\rm{HI}}$, following the increasing bias of the cosmic HII regions.
As is the case for 21cm, the strongest cross signal occurs when the IGM is
colder than the CMB, providing a large contrast between the neutral regions and
the ionized regions which host LAEs. We also vary the topology of reionization
and the epoch of X-ray heating. The cross-correlation during the first half of
reionization is sensitive to these topologies, and could thus be used to
constrain them.
