  We study particle production and the corresponding entropy increase in the
context of cosmology with dynamical vacuum. We focus on the particular form
that has been called "running vacuum model" (RVM), which is known to furnish a
successful description of the overall current observations at a competitive
level with the concordance $\Lambda$CDM model. It also provides an elegant
global explanation of the cosmic history from a non-singular initial state in
the very early universe up to our days and further into the final de Sitter
era. The model has no horizon problem and provides an alternative explanation
for the early inflation and its graceful exit, as well as a powerful mechanism
for generating the large entropy of the current universe. The energy-momentum
tensor of matter is generally non-conserved in such context owing to particle
creation or annihilation. We analyze general thermodynamical aspects of
particle and entropy production in the RVM. We first study the entropy of
particles in the comoving volume during the early universe and late universe.
Then, in order to obtain a more physical interpretation, we pay attention to
the entropy contribution from the cosmological apparent horizon, its interior
and its surface. On combining the inner volume entropy with the entropy on the
horizon, we elucidate with detailed calculations whether the evolution of the
entropy of the RVM universe satisfies the Generalized Second Law of
Thermodynamics. We find it is so and we prove that the essential reason for it
is the existence of a positive cosmological constant.
