Carbon burning cannot explain puffy hypervelocity white dwarfs

Several hypervelocity white dwarfs (HVWDs) with space velocities of $\gtrsim 1000\,\mathrm{kms}^{-1}$ have recently been discovered. One possible origin of these stars is the dynamically-driven double-degenerate double-detonation (D6) scenario, in which an accreting sub-Chandrasekhar mass carbon-oxygen (CO) WD detonates as a SN Ia. In this scenario, the less massive WD may survive its companion's detonation and be ejected as a HVWD. Most of the observed HVWDs are hotter and puffier than normal WDs, perhaps due to their recent proximity to a SN. In this work, we test whether these properties can be explained by long-lived stable carbon (C) burning in the interiors of CO WD donors triggered by a SN shock. We model the long-term evolution of CO WDs following rapid energy injection using 1D models. We find that stable C burning can be ignited in CO WDs with masses of $0.95 - 1.10\,M_{\odot}$ if SN energy penetrates sufficiently deeply. The resulting born-again stars settle on the C-burning main sequence while they convert their interiors from C and O to Ne and Mg, where they have temperatures and radii similar to some of the observed HVWDs. However, the timescale over which C-burning WDs remain inflated is $\lesssim 10^5\,$yr, which is at least an order of magnitude shorter than the kinematic ages of observed hot HVWDs. We conclude that observed HVWDs are unlikely to be inflated by C burning. The stellar evolution of observed HVWDs remains an open problem.