We study the equation of state for neutron matter at finite temperature
based on low-momentum nucleon-nucleon and three-nucleon interactions. The
free energy is obtained using a loop expansion around the Hartree-Fock
energy, and various thermodynamic properties are derived.
The perturbative expansion includes the contributions from second-order
anomalous and normal diagrams. By studying the cutoff dependence,
we provide an estimate for the theoretical error of our results,
important for extrapolations to astrophysical conditions in core-collapse
supernovae and neutron star mergers. We compare our results to the
model-independent virial equation of state and to variational calculations.