Holographic Cosmology from the First Law of Thermodynamics and the Generalized Uncertainty Principle

The cosmological Friedmann equation sourced by the trace anomaly of a conformal field theory that is dual to the five-dimensional Schwarzschild-AdS geometry can be derived from the first law of thermodynamics if the apparent horizon of the boundary spacetime acquires a logarithmically-corrected Bekenstein-Hawking entropy. It is shown that such a correction to the entropy can arise when the generalized uncertainty principle (GUP) is invoked. The necessary condition for such a thermodynamic derivation directly relates the GUP parameter to the conformal anomaly. It is consistent with the existence of a gravitational cutoff for a theory containing $n$ light species. The absolute minimum in position uncertainty can be identified with the scale at which gravity becomes effectively five-dimensional.