  The relative heat content associated with a subset $\Omega\subset M$ of a
sub-Riemannian manifold, is defined as the total amount of heat contained in
$\Omega$ at time $t$, with uniform initial condition on $\Omega$, allowing the
heat to flow outside the domain. In this work, we obtain a fourth-order
asymptotic expansion in square root of $t$ of the relative heat content
associated with relatively compact non-characteristic domains. Compared to the
classical heat content that we studied in [Rizzi, Rossi - J. Math. Pur. Appl.,
2021], several difficulties emerge due to the absence of Dirichlet conditions
at the boundary of the domain. To overcome this lack of information, we combine
a rough asymptotic for the temperature function at the boundary, coupled with
stochastic completeness of the heat semi-group. Our technique applies to any
(possibly rank-varying) sub-Riemannian manifold that is globally doubling and
satisfies a global weak Poincar\'e inequality, including in particular
sub-Riemannian structures on compact manifolds and Carnot groups.
