  General relativistic bending of light dramatically alters the variability of
X-ray emission originating from the surfaces of ultramagnetic neutron stars. We
construct radiative equilibrium models of such strongly magnetic cooling
neutron stars with light-element atmospheres to compute the angle- and
energy-dependent intensity emerging from their surfaces and find that the
beaming of surface emission is predominantly non-radial. The combination of
this radiation pattern with the calculations of light bending yields pulse
amplitudes that vary non-monotonically with the neutron star compactness and
the size of the emitting region. The significant suppression of the pulse
amplitude for large emitting areas provides very strong constraints on the
mechanisms that can simultaneously produce high periodic variability and X-ray
luminosity. We apply these results to the thermally-emitting magnetar models of
anomalous X-ray pulsars (AXPs), which are bright slowly-rotating X-ray sources
with large pulse amplitudes. We use the observed fluxes and pulse amplitudes
for all known AXPs and show that thermal emission from two antipodal regions on
their surfaces, as predicted by some magnetar models, is inconsistent with
these observed properties.
