Entropy of matter on the Carroll geometry

Two prescriptions for the construction of Carroll geometries, the expansion of geometric variables near horizon and the expansion of metric with zero limit of the expansion parameter $c$ (speed of light in vacuum), are known to complement each other. The entropy of an ideal gas, confined in a box and kept very close to the horizon, depends on the transverse area of the container. We show this by using the Carroll geometry constructed through the expansion of the metric and then taking the zero limit of the expansion parameter $c$. Therefore, the present analysis assures the complementing nature of two ways of finding the Carroll geometry from the thermodynamical point of view.