Horizon-entropy increase laws for spherically symmetric horizons in Brans-Dicke theory

We derive the horizon-entropy increase law for spherically symmetric quasi-local horizons in Brans-Dicke theory. The quasi-local horizons used do not marginally trap null rays, and hence are not apparent horizons or foliated by marginally trapped surfaces, but instead have instantaneously constant gravitational entropy in the outgoing null direction. The relation derived has a very direct comparison with the horizon-entropy increase law for event horizon