Brans-Dicke theory and thermodynamical laws on apparent and event horizons

In this work, we describe the Brans-Dicke (BD) theory of gravity and give a particular solution by choosing a power law form for the scalar field φ and constant ω. If we assume that the first law of thermodynamics and the entropy formula hold on the apparent horizon, then we recover the Friedmann equations. Next, assuming the first law of thermodynamics holds, the validity conditions of the generalized second law (GSL) on the event horizon are presented. If we impose the entropy relation on the horizon, without using the first law, then we also obtain the validity condition of the GSL on the event horizon. The validity of the GSL completely depends on the BD model scalar field solutions. We have justified that the two processes are equivalent on the apparent horizon, but on the event horizon they are not equivalent. If the first law is valid on the event horizon, then the GSL may be satisfied by the BD solution, but if the first law is not satisfied then the GSL is not satisfied by the BD solution. Therefore, the first law always favours the GSL on the event horizon. In our effective approach, the first law and the GSL are always satisfied on the apparent horizon, which does not depend on the BD theory of gravity. (author)