  It is expected that the cosmological black holes are the closest realistic
solutions of gravitational theories and they evolve with time. Moreover, the
natural way of defining thermodynamic entities for the stationary ones is not
applicable in the case of a time dependent spacetime. Here we confine our
discussion within the Sultana-Dyer metric which is a cosmological black hole
solution of Einstein's gravity. In literature, there exists two expressions of
horizon temperature -- one is time dependent and the other does not depend on
time. To single out the correct one we find the temperature by studying the
Hawking effect in the tunnelling formalism. This leads to time dependent
structure. After identifying the correct one, the Einstein's equations are
written on the horizon and we show that this leads to the first law of
thermodynamics. In this process the expressions for horizon entropy and energy,
obtained earlier by explicit calculations, are being used. This provides the
evidence that Einstein's equations have thermodynamic structure even for a
cosmological black hole spacetime. Moreover, this study further clarifies the
correctness of the expressions for the thermodynamic quantities; like
temperature, entropy and internal energy.
