If I understand this correctly, the authors find a correlation between free energy and stability. Deviations from the expected (linear?) relationship may represent biological constraints (possibly function??).
Conversely, one also might posit that the usages of codons reflect biologically relevant features of those DNA sequences containing a statistical overabundance of energetically favorable or unfavorable codons. The altered energy profiles of such DNA sequences relative to a statistically expected distribution of codons/energies may reflect the existence of biological constraints that do not apply to an average sequence. In other words, codon usage that deviates from the average expected distribution (either positive or negative) may reflect altered biological constraints.