In this work we explore the scope of two holographic dark energy models
within the interacting scenario for the dark sector with spatial curvature. As
first holographic model we consider the usual formula for the dark energy
density and the Hubble scale. For the second model we have a function in the
holographic formula instead the usual constant parameter. In this description
both holographic models admit a future singularity. The use of recent
cosmological data is considered only to support the physical results of our
proposal. With the use of the best fit values of the cosmological parameters we
show that the interaction term for each holographic model, $Q$, keeps positive
along the cosmic evolution. The form of the interaction term is not given a
priori, is reconstructed from the model. The temperatures for the components of
the dark sector are computed and exhibit a growing behavior in both scenarios.
The cosmic evolution in this context it is not adiabatic and the second law it
is fulfilled only under certain well-established conditions for the
temperatures of the cosmic components and positive $Q$-term.
