  While standing fast sausage modes in flare loops are often invoked to
interpret quasi-periodic pulsations (QPPs) in solar flares, it is unclear as to
how they are influenced by the combined effects of a continuous transverse
structuring and a finite internal plasma beta ($\beta_{\rm i}$). We derive a
generic dispersion relation (DR) governing linear sausage waves in straight
magnetic tubes for which plasma pressure is not negligible and the density and
temperature inhomogeneities of essentially arbitrary form take place in a layer
of arbitrary width. Focusing on fast modes, we find that $\beta_{\rm i}$ only
weakly influences $k_{\rm c}$, the critical longitudinal wavenumber separating
the leaky from trapped modes. Likewise, for both trapped and leaky modes, the
periods $P$ in units of the transverse fast time depend only weakly on
$\beta_{\rm i}$, which is compatible with the fact that the effective
wavevectors of fast sausage modes are largely perpendicular to the background
magnetic field. However, a weak $\beta_{\rm i}$ dependence of the damping times
$\tau$ is seen only when the length-to-radius ratio $L/R$ is $\sim 50\%$ larger
than some critical value $\pi/(k_{\rm c} R)$, which itself rather sensitively
depends on the density contrast, profile steepness as well as on how the
transverse structuring is described. In the context of QPPs, we conclude that
the much simpler zero-beta theory can be employed for trapped modes, as long as
one sees the deduced internal Alfv\'en speed as actually being the fast speed.
In contrast, effects due to a finite beta in flare loops should be considered
when leaky modes are exploited.
