Assuming that an accelerated observer with four-velocity (Formula Presented) in a curved spacetime attributes the standard Bekenstein–Hawking entropy and Unruh temperature to his “local Rindler horizon”, I show that the change in horizon area under parametric displacements of the horizon has a very specific thermodynamic structure. Specifically, it entails information about the time–time component of the Einstein tensor: (Formula Presented). Demanding that the result holds for all accelerated observers, this actually becomes a statement about the full Einstein tensor, (Formula Presented). I also present some perspectives on the free fall with four-velocity (Formula Presented) across the horizon that leads to such a loss of entropy for an accelerated observer. Motivated by results for some simple quantum systems at finite temperature T, we conjecture that at high temperatures, there exists a universal, system-independent curvature correction to partition function and thermal entropy of any freely falling system, characterised by the dimensionless quantity (Formula Presented). © 2017, Springer International Publishing AG.