One of the canonical approaches to quantum gravity is quantum geometrodynamics. It arises from the quantization of Hamiltonian constraints. This method gives rise to the Wheeler–DeWitt equation, which can be solved in minisuperspace models to obtain the wavefunction of the universe. Despite its long history, this approach still faces fundamental challenges: operator ordering ambiguity, the problem of time, and the lack of a clear probabilistic interpretation remain open issues. In our work, we discuss how studies of entropy may shed some light on these difficulties.

We study a toy model of the Wheeler–DeWitt equation for a Friedmann–Lemaître–Robertson–Walker (FLRW) universe with a non-zero cosmological constant and matter content consisting of a massive scalar field coupled to gravity. We construct the universal wavefunction using a spectral expansion method, which allows us to obtain numerical solutions on the entire domain while avoiding instabilities. This approach enables a detailed analysis of the wavefunction and provides access to both gravitational and matter degrees of freedom without the use of the slow-roll approximation.

Based on the obtained universal wavefunction, we discuss possible entropy measures and their implications, as well as the potential role of matter–geometry entanglement as an indicator of the quantum-to-classical transition. We conclude with a comment on the use of entropy-related quantities as arrows of time.