Recent discovery of a nonclassical rotational inertia (NCRI) in solid He-4 below 0.2 K by Kim and Chan has revived great interest in the problem of supersolidity and initiated intensive study on the properties of solid He-4. A direct proof that the onset of NCRI corresponds to the supersolid transition would be the observation of a corresponding drop of the entropy of solid He-4 below the transition temperature. We have measured the melting pressure of ultrapure He-4 in the temperature range from 0.01 to 0.45 K with several single crystals grown at different pressures and with the accuracy of 0.5 mu bar. In addition, supplementary measurements of the pressure in liquid He-4 at constant volume have been performed, which allowed us to eliminate the contribution of the temperature-dependent properties of the pressure gauge from the measured melting pressure data. With the correction to the temperature-dependent sensitivity of the pressure gauge, the variation of the melting pressure of He-4 below 320 mK obeys the pure T-4 law due to phonons with the accuracy of 0.5 mu bar, and no sign of the transition is seen (Todoshchenko et al. in JETP Lett. 85:454, 2007). This sets the upper limit of similar to 5 center dot 10(-8) R for a possible excess entropy in high-quality He-4 crystals below 320 mK. At higher temperatures the contribution from rotons in the superfluid He-4 has been observed. The thermal expansion coefficient of the superfluid He-4 has been measured in the range from 0.01 to 0.7 K with the accuracy of similar to 10(-7) 1/K, or by two orders of magnitude better than in previous measurements. The roton contributions to the melting pressure and to the pressure in liquid at a constant volume are consistent and yield the value of 6.8 K for the roton gap, which is very close to the values obtained with other methods. As no contribution due to weakly interacting vacancies to the melting pressure of He-4 has been observed, the lower limit of about 5.5 K for their activation energy can be set.