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The thermal conductivity κ, heat capacity per unit volume ρc_p and glass transition behaviour under pressure have been established for medium and high vinyl content polybutadiene PB with molecular weights 2600 and 100 000 and their highly cross-linked (ebonite) states obtained purely by high-pressure high-temperature treatments. Cross-linking eliminates the glass transitions and increases κ by as much as 50% at 295 K and 1 atm, and decreases ρc_p to a limiting level close to that of the glassy state of PB, which is reached before the ultimate cross-link density is achieved. The pressure and temperature behaviours of κ are strongly changed by cross-links, which increases the effect of temperature but decreases the effect of pressure. We attribute these changes to a cross-linked induced permanent densification and consequential increase of phonon velocity simultaneously as conduction along polymer chains is disrupted. The glass transition temperatures for a time scale of 1 s are described to within 0.5 K by: T_g(p) = 202.5 (1 + 2.94 p)^0.286 and T_g(p) = 272.3 (1 + 2.57 p)^0.233 (p in GPa and T in K) up to 1 GPa, for PB2600 and PB100000, respectively, and can be estimated for medium and high vinyl content PBs with molecular weights in between by a constant, pressure independent, shift in temperature.