We calculate the equation of state (EOS) of asymmetric nuclear matter at finite temperatures with the cluster variational method based on the realistic nuclear Hamiltonian composed of the AV18 and UIX nuclear potentials. The free energy is calculated with an extension of the variational method proposed by Schmidt and Pandharipande. The obtained thermodynamic quantities such as entropy, internal energy, pressure and chemical potential derived from the free energy are reasonable. It is also found that the present variational calculation is self-consistent. These thermodynamic quantities are essential ingredients in our project for constructing a new nuclear EOS applicable to supernova simulations.