Thermal properties of low-density neutron matter are investigated by nuclear many-body calculation on three-dimensional cubic lattice. Determinantal quantum　Monte Carlo calculations are carried out for the Hamiltonian determined by the use of the method of effective field theory. The investigation is an ongoing project, and an overview of the project will be presented, including its future prospects. The present finding will be also discussed on thermal properties of neutron-neutron correlations in very low density neutron matter: a phase transition from a superfluid state to normal, through a crossover region of Bose-Einstein condensation.