One problem with this argument is the earth is not a
closed system. Another is, biological complexity is not the same type
as any sort of complexity that would be relevant when talking about
thermodynamics. Allow me to explain.

The second law of thermodynamics is not in conflict with
the theory of evolution. The sun pumps out a lot of energy. A small
percentage of this energy is used by plants to make sugar via
photosynthesis. From there, energy flows through the food web.
For the solar system there is a net increase in entropy (and it's
not even closed).

No extra energy other than that derived from photosynthesis is needed
for populations to evolve. Evolution is a change in the gene pool of
a population over time. Several mechanisms can bring about evolutionary
change, they are: natural selection, genetic drift, mutation,
recombination and gene flow.

 Natural selection is differential reproductive success of
genetic variants in the gene pool. This requires no "extra" energy
other than the energy the organisms obtain from photosynthesis
(either directly or indirectly).

 Genetic drift is a binomial sampling error of the gene
pool. In other words, each generation is a sample of genes from
the previous generation's gene pool. In a finite population, gene
frequencies change slightly each generation due to chance. Again,
no "extra" energy is needed to fuel evolution.

 Mutation is a change in a gene. This can be caused by
DNA replication errors or by damage due to radiation. Once again,
no conflict with the second law of thermo (although the energy
to cause the mutation does not come from photosythesis in the
case of radiation damage.)

 Recombination is exchange of genetic information
(usually) with sister chromatids or homologous chromosomes. The
enzymes that accomplish this work do so with energy derived from
photosynthesis (again, either directly or indirectly). Once again,
no conflict with thermodynamics.

 And finally gene flow is migration of new genes into a
population. This often occurs by fertile hybrids "bridging"
two gene pools. No thermodynamic problems here.