  By means of thermal expansion and specific heat measurements on the
high-pressure phase of (VO)$_2$P$_2$O$_7$, the effects of two energy scales of
the weakly dimerized antiferromagnetic $S$ = 1/2 Heisenberg chain are explored.
The low energy scale, given by the spin gap $\Delta$, is found to manifest
itself in a pronounced thermal expansion anomaly. A quantitative analysis,
employing T-DMRG calculations, shows that this feature originates from changes
in the magnetic entropy with respect to $\Delta$, $\partial S^{m}/ \partial
\Delta$. This term, inaccessible by specific heat, is visible only in the
weak-dimerization limit where it reflects peculiarities of the excitation
spectrum and its sensitivity to variations in $\Delta$.
