  We investigate the finite-temperature properties of attractive
three-component (colors) fermionic atoms in optical lattices using a
self-energy functional approach. As the strength of the attractive interaction
increases in the low-temperature region, we observe a second-order transition
from a Fermi liquid to a color superfluid (CSF), where atoms from two of the
three colors form Cooper pairs. In the strong attractive region, we observe a
first-order transition from a CSF to a trionic state, where three atoms with
different colors form singlet bound states. A crossover between a Fermi liquid
and a trionic state is observed in the high-temperature region. We present a
phase diagram covering zero to finite temperatures. We demonstrate that the CSF
transition temperature is enhanced by the anisotropy of the attractive
interaction.
