Temperature anisotropy has been frequently observed in the solar corona and the solar wind, yet poorly represented in computational models of the solar wind. Therefore, we have included proton temperature anisotropy in our Alfvén wave solar model (AWSoM). This model solves the magnetohydrodynamic equations augmented with low-frequency Alfvén wave turbulence. The wave reflection due to Alfvén speed gradient and field-aligned vorticity results in turbulent cascade. At the gyroradius scales, the apportioning of the turbulence dissipation into coronal heating of the protons and electrons is through stochastic heating. This paper focuses on the impacts of the proton temperature anisotropy on the solar wind. We apply AWSoM to simulate the steady solar wind from the corona to 1 AU using synoptic magnetograms. The Alfvén wave energy density at the inner boundary is prescribed with a uniform Poynting flux per field strength. We present the proton temperature anisotropy distribution, and investigate the firehose instability in the heliosphere from our simulations. In particular, the comparisons between the simulated and observed solar wind properties at 1 AU during the ramping-up phase and the maximum of solar cycle 24 imply the importance of addressing the proton temperature anisotropy in solar wind modelling to capture the fast solar wind speed.