  Recent theoretical studies [Chen et al., Phys. Rev. B 82, 174440 (2010),
Ishizuka et al., Phys. Rev. B 90, 184422 (2014)] for the magnetic Mott
insulator Ba2NaOsO6 have proposed a low-temperature order parameter that breaks
lattice rotational symmetry without breaking time reversal symmetry leading to
a nematic phase just above magnetic ordering temperature. We present
high-resolution calorimetric and magnetization data of the same Ba2NaOsO6
single crystal and show evidence for a weakly field-dependent phase transition
occurring at a temperature of Ts ~ 9.5K, above the magnetic ordering
temperature of Tc ~ 7.5K. This transition appears as a broadened step in the
low-field temperature dependence of the specific heat. The evolution of the
phase boundary with applied magnetic field suggests that this phase coincides
with the phase of broken local point symmetry seen in high field NMR
experiments [Lu et al., Nat.\ Comm. 8 14407 (2017)]. Furthermore, the magnetic
field dependence of the specific heat provides clear indications for magnetic
correlations persisting at temperatures between Tc and Ts where long-range
magnetic order is absent giving support for the existence of the proposed
nematic phase.
