  We discuss universal features on the electron spin resonance (ESR) of a
temperature-induced Tomonaga-Luttinger liquid phase in a wide class of weakly
coupled $S=1/2$ antiferromagnetic spin chains such as spin ladders, spin tubes
and three-dimensionally coupled spin chains. We show that the ESR linewidth of
various coupled chains increases with lowering temperature while the linewidth
of a single spin chain is typically proportional to temperature. This
broadening with lowering temperature is attributed to anisotropic interchain
interactions and has been indeed observed in several kinds of three-dimensional
(3D) magnets of weakly coupled spin chains above the 3D ordering temperature.
We demonstrate that our theory can account for anomalous behaviors of the
linewidths in an $S=1/2$ four-leg spin tube compound Cu$_2$Cl$_4 \cdot
$H$_8$C$_4$SO$_2$ (abbreviated to Sul-Cu$_2$Cl$_4$) and a three-dimensionally
coupled $S=1/2$ spin chain compound CuCl$_2\cdot 2$NC$_5$H$_5$.
