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We develop the Brans-Dicke theory of gravity in the context of varyingconstants of Nature. Using the unimodular formalism of General Relativity, wecreate a platform to provide physical relational times giving the evolution ofphysical constants. We therefore review the ideas and experiments behindvarying constants, mostly focusing on the speed of light and the gravitationalconstant. Then, we apply this idea to the energy conservation in cosmology,illustrating the arising patterns. Motivated by a varying gravitationalconstant resulting from Mach's principle, we develop the unimodular formalismof varying constants in the Brans-Dicke theory. Doing so, we obtain severaloriginal results, some of which can be compared with phenomenologicalobservation. Finally, we suggest how a varying Brans-Dicke parameter could belinked to the Cosmological Constant problem.

Varying Constants and the Brans-Dicke theory: a new landscape in cosmological energy conservation