  We present the study of gas phases around cosmic-web filaments detected in
the TNG300-1 hydro-dynamical simulation at redshift z=0. We separate the gas in
five different phases according to temperature and density. We show that
filaments are essentially dominated by gas in the warm-hot intergalactic medium
(WHIM), which accounts for more than 80% of the baryon budget at $r \sim 1$
Mpc. Apart from WHIM gas, cores of filaments ($r<1$ Mpc) also host large
contributions other hotter and denser gas phases, whose fractions depend on the
filament population. By building temperature and pressure profiles, we find
that gas in filaments is isothermal up to $r \sim 1.5$ Mpc, with average
temperatures of T_core = $4-13 \times 10^5$ K, depending on the large scale
environment. Pressure at cores of filaments is on average P_core = $4-12 \times
10^{-7}$ keV/cm^3, which is ~1000 times lower than pressure measured in
observed clusters. We also estimate that the observed Sunyaev-Zel'dovich (SZ)
signal from cores of filaments should range between $0.5 < y < 4.1 \times
10^{-8}$, and these results are compared with recent observations. Our findings
show that the state of the gas in filaments depend on the presence of haloes,
and on the large scale environment.
