On the foundations of entropic cosmologies: inconsistencies, possible solutions and dead end signs

In this letter we explore the foundations of entropic cosmology and highlight some important flaws which have emerged and adopted in the recent literature. We argue that, when applying entropy and temperature on the cosmological horizon by assuming the holographic principle for all thermodynamic approaches to cosmology and gravity, one must derive the consistent thermodynamic quantities following Clausius relation. One key assumption which is generally overlooked, is that in this process one must assume a mass-to-horizon relation, which is generally taken as a linear one. We show that, regardless of the type of entropy chosen on the cosmological horizon, when a thermodynamically consistent corresponding temperature is considered, all modified entropic force models are equivalent to and indistinguishable from the original entropic force models based on standard Bekenstein entropy and Hawking temperature. As such, they are also plagued by the same problems and inability to describe in a satisfactory qualitative and quantitative way the cosmological dynamics as it emerges from the probes we have. We also show that the standard accepted parameterization for Hawking temperature (including a $\gamma$ rescaling) is actually not correctly applied, namely, it is not related to entropy in a thermodynamically consistent way. Finally, we clearly state that the explicit form of the entropic force on cosmological horizons is mostly dictated by the assumption on the mass-to-horizon relation. As such, we discuss what should be done in order to fix all such issues, and what conceptually could be implied by its correct implementation in order to advance in the field.