  We establish a general lower bound for the entropy production rate based on
the Kullback-Leibler divergence and the Logarithmic-Sobolev constant that
characterizes the time-scale of relaxation. This bound can be considered as an
enhanced second law of thermodynamics. As a crucial application, we find a
universal trade-off relation between the dissipation rate and relaxation
timescale in thermal relaxation. Importantly, we show that a thermodynamic
upper bound on the relaxation time between two given states follows directly
from the trade-off relation, serving as an inverse speed limit throughout the
entire time region. Our findings unveil some hidden universal behaviors of
thermal relaxation processes, which can also be extended to open quantum
systems.
