  Cosmological hydrodynamical simulations predict that a large fraction of all
baryons reside within mildly non-linear structures with temperatures in the
range $10^{5}-10^{7}$K. As the gas is highly ionized, it could be detected by
the temperature anisotropies generated on the Cosmic Microwave Background
radiation. We refine our previous estimates of the thermal Sunyaev-Zeldovich
effect by introducing a non-polytropic equation of state to model the
temperature distribution of the shock heated gas derived from
temperature-density phase diagrams of different hydrodynamical simulations.
Depending on the specific model, the Comptonization parameter varies in the
range $10^{-7}\le y_c \le 2\times 10^{-6}$, compatible with the FIRAS upper
limit. This amplitude is in agreement with a simple toy model constructed to
estimate the average effect induced by filaments of ionized gas. Using the
log-normal probability density function we calculate the correlation function
and the power spectrum of the temperature anisotropies generated by the WHIM
filaments. For a wide range of the parameter space, the maximum amplitude of
the radiation power spectrum is $(\ell+1)\ell C_{\ell}/2\pi =0.7-70(\mu K)^2$
at $\ell \approx 200-500$. This amplitude scales with baryon density, Hubble
constant and the amplitude of the matter power spectrum $\sigma_8$ as
$[(\ell+1)\ell C_\ell]_{\mathrm{max}}/2\pi \propto \sigma_8^{2.6}(\Omega_b
h)^2$. Since the thermal Sunyaev-Zeldovich effect has a specific frequency
dependence, we analyze the possibility of detecting this component with the
forthcoming Planck data.
