  The angular power spectrum of the thermal Sunyaev-Zel'dovich (tSZ) effect is
highly sensitive to cosmological parameters such as sigma_8 and Omega_m, but
its use as a precision cosmological probe is hindered by the astrophysical
uncertainties in modeling the gas pressure profile in galaxy groups and
clusters. In this paper we assume that the relevant cosmological parameters are
accurately known and explore the ability of current and future tSZ power
spectrum measurements to constrain the intracluster gas pressure or the
evolution of the gas mass fraction, f_gas. We use the CMB bandpower
measurements from the South Pole Telescope and a Bayesian Markov Chain Monte
Carlo (MCMC) method to quantify deviations from the standard, universal gas
pressure model. We explore analytical model extensions that bring the
predictions for the tSZ power into agreement with experimental data. We find
that a steeper pressure profile in the cluster outskirts or an evolving f_gas
have mild-to-severe conflicts with experimental data or simulations. Varying
more than one parameter in the pressure model leads to strong degeneracies that
cannot be broken with current observational constraints. We use simulated
bandpowers from future tSZ survey experiments, in particular a possible 2000
deg^2 CCAT survey, to show that future observations can provide almost an order
of magnitude better precision on the same model parameters. This will allow us
to break the current parameter degeneracies and place simultaneous constraints
on the gas pressure profile and its redshift evolution, for example.
