  We explore the thermal and magnetic-field structure of a late-stage
proto-neutron star. We find the dominant contribution to the entropy in
different regions of the star, from which we build a simplified equation of
state for the hot neutron star. With this, we numerically solve the stellar
equilibrium equations to find a range of models, including magnetic fields and
rotation up to Keplerian velocity. We approximate the equation of state as a
barotrope, and discuss the validity of this assumption. For fixed
magnetic-field strength, the induced ellipticity increases with temperature; we
give quantitative formulae for this. The Keplerian velocity is considerably
lower for hotter stars, which may set a de-facto maximum rotation rate for
non-recycled NSs well below 1 kHz. Magnetic fields stronger than around
$10^{14}$ G have qualitatively similar equilibrium states in both hot and cold
neutron stars, with large-scale simple structure and the poloidal field
component dominating over the toroidal one; we argue this result may be
universal. We show that truncating magnetic-field solutions at low multipoles
leads to serious inaccuracies, especially for models with rapid rotation or a
strong toroidal-field component.
