This is a well-studied problem. The effect of rotation on the structure of a low-ish mass star (like the Sun) is summarised by Eggenberger (2013).
Such stars are never observed to rotate so fast that the rotation plays any significant role in their hydrostatic equilibrium, however rotation does play a role by causing additional mixing in the star.
This is important for two reasons: (i) it inhibits the gradual diffusion of helium towards the core, this slightly lowers the opacity in the core and raises it in the envelope (compared with a non-rotating star). This results in a slightly higher luminosity and a slightly higher surface temperature. (ii) More importantly, the additional mixing brings additional hydrogen into the core and this increases the lifetime on the main sequence.
The effects on stars of a solar mass are however unlikely to be very significant in practice, because these stars efficiently lose angular momentum via a magnetised wind during their early lives and the effects of rotation are unlikely to be significant even at several times the solar rotation rate.
The effects on more massive stars can be more severe. These can spin at an appreciable fraction of their break-up rate for a large part of their lives and do not lose angular momentum as efficiently as lower mass stars (they do not have magnetised winds). The effects are described in the canonical paper of Meynet & Maeder (2000); they are more pronounced than for lower mass stars and more complicated because of the radiative envelope and uncertainties in the rotation-dependence of the significant mass-loss.
The hydrostatic effects of rotation are expected to be important early in the main sequence and contribute to a slightly lower surface temperature. At later times the dominant effects are caused by changes in the mixing and diffusion near the core and in the envelope as for the lower mass stars, resulting in higher luminosities and hotter temperatures. The main sequence lifetimes can be extended by 30% due to the additional mixing of fresh hydrogen fuel into the core.