Synchrotron phase nano-CT is a very useful technique for studying bone diseases, which requires investigating bone at the cellular level. Nevertheless, imaging biological tissue at this resolution is challenging due to the very high radiation dose. Compressed sensing provides a framework that permits to reconstruct an image from a limited amount of data. The most promising way to reduce radiation exposure in X-ray CT is to reduce the number of projections. The aim of this study is to assess the use of compressed sensing to reduce dose for synchrotron phase nano-CT for bone applications. In this paper, we address the tomographic reconstruction step by posing a total variation problem and solving it with the Split Bregman formulation. To assess the proposed method we created different low-dose imaging scenarios by reducing the number of projections, and tested them on several bone samples. The proposed method allowed accurate reconstruction using 1/4th of the projections, preserving bone features, details, and a high signal to noise ratio.