To investigate the effects of local density fluctuations on phonon propagation in a hydrogen bonded structure, we studied the thermal conductivity kappa of the crystal, liquid, and glassy states of pure glycerol as a function of the temperature, T, and the pressure, p. We find that the following: (i) K-crystal is 3.6-times the kappa(liquid) value at 140 K at 0.1 MPa and 2.2-times at 290 K, and it varies with T according to 138 x T-0.95; (ii) the ratio kappa(liquid) (P)/kappa(liquid) (0.1 MPa) is 1.45 GPa(-1) at 280 K, which, unexpectedly, is about the same as kappa(crystal) (p)/kappa(crystal) (0.1 MPa) of 1.42 GPa(-1) at 298 K; (iii) kappa(glass) is relatively insensitive to T but sensitive to the applied p (1.38 GPa(-1) at 150 K); (iv) kappa(glass)-T plots show an enhanced, pressure-dependent peak-like feature, which is due to the glass to liquid transition on heating; (v) continuous heating cold-crystallizes ultraviscous glycerol under pressure, at a higher T when p is high; and (vi) glycerol formed by cooling at a high p and then measured at a low p has a significantly higher kappa than the glass formed by cooling at a low p. On heating at a fixed low p, its kappa decreases before its glass-liquid transition range at that p is reached. We attribute this effect to thermally assisted loss of the configurational and vibrational instabilities of a glass formed at high p and recovered at low p, which is different from the usual glass-aging effect. While the heat capacity, entropy, and volume of glycerol crystal are less than those for its glass and liquid, kappa(crystal) of glycerol, like its elastic modulus and refractive index, is higher. We discuss these findings in terms of the role of fluctuations in local density and structure, and the relations between K and the thermodynamic quantities. (C) 2016 AIP Publishing LLC.