A possible approach to reducing Nb<sub>3</sub>Sn magnet training is to increase the energy margin of Nb<sub>3</sub>Sn conductors by enhancing their specific heat (C<sub>p</sub>). For this study, we have been developing Nb<sub>3</sub>Sn conductors with increased C<sub>p</sub> by incorporating substances with high C<sub>p</sub> at 2-10 K based on a conductor design that is compatible with standard Nb<sub>3</sub>Sn strand production. In the past couple of years our efforts have been mainly focused on improving strand design (e.g., position of high- C<sub>p</sub> filaments, thickness of the Cu tube for the high-C<sub>p</sub> filaments, ratio of the Cu powder to the high-C<sub>p</sub> substance, filament spacing, etc.) in order to obtain good strand drawability and to reduce degradation after rolling, which is needed for production of Rutherford cables. We also tried a new high-C<sub>p</sub> substance, Gd<sub>2</sub>O<sub>2</sub>S, and verified that it has much higher C<sub>p</sub> over the whole magnetic field range than the Gd<sub>2</sub>O<sub>3</sub> we used before. With that development work we can now produce high-C<sub>p</sub> strands with good drawability and low levels of degradation after rolling. This paper reports our findings and the current status of the development of high-C<sub>p</sub> Nb<sub>3</sub>Sn conductors.