What is the definition of thermodynamic stability? The term thermodynamic stability is used on this site, but I can't find a good definition. Is is a quantitative or a qualitative concept? Does it apply to a single compound, or a pair of compounds? Does it a relative or an absolute value?
An answer to another question, Thermodynamic stability of benzene derivatives , states

Thermodynamic stability of compounds can be determined by obviously enthalpy of formation ($\Delta H_{_\mathrm f}$) of individual compounds. The enthalpy of formation will be lesser if the compound is formed from its constituent elements enjoys some greater stability.

The enthalpy of formation considers the synthesis of a given compound from the elements. If the enthalpy of formation of two substances like bromomethane and chloromethane are different, it might be due to differences in bond strengths (or electronic states) in the two compounds, or differences in bond strengths (or electronic states) comparing elemental bromine and elemental chlorine.
I do understand the concept of thermodynamically favored product. In this case, there are competing reactions leading to different products. Because they start with the same reactants, there is no issue about the reference state. However,  if I compare two substance with different composition (i.e. not isomers), I don't know what it means to compare their thermodynamic stability. I also understand local and global minima, as discussed in this answer which references a single set of atoms that undergoes transformation. The problem arises when comparing compounds with different sets of atoms.
Thermodynamics usually is pretty well defined. Thermodynamic stability, however, is a bit of a mystery to me.
The full phrase should be thermodynamic stability with respect to ____, where the dash indicates a process, or a chemical reaction.
A mixture of hydrogen and oxygen is thermodynamically unstable with respect to water formation.
Similarly, a diamond is not forever (which may not please De Beers and ladies). It is thermodynamically unstable with respect to conversion to graphite.
Also, thermodynamic stability is a relative term which is often contrasted with reactivity or kinetic stability.
Diamond is kinetically stable at room temperature for the same process (lucky ladies can smile again).