  The first excited $J^\pi=0^+$ state of $^{12}$C, the so-called Hoyle state,
plays an essential role in a triple-$\alpha$ ($^4$He) reaction, which is a main
contributor to the synthesis of $^{12}$C in a burning star. We investigate the
Coulomb screening effects on the energy shift of the Hoyle state in a thermal
plasma environment using precise three-$\alpha$ model calculations. The Coulomb
screening effect between $\alpha$ clusters are taken into account within the
Debye-H\"uckel approximation. To generalize our study, we utilize two standard
$\alpha$-cluster models, which treat the Pauli principle between the $\alpha$
particles differently. We find that the energy shifts do not depend on these
models and follow a simple estimation in the zero-size limit of the Hoyle state
when the Coulomb screening length is as large as a value typical of such a
plasma consisting of electrons and $\alpha$ particles.
