The electromagnetic proton firehose instability may grow in a plasma if the proton velocity distribution is approximately bi-Maxwellian and T∥p>T⊥p, where the directional subscripts denote directions relative to the background magnetic field. Linear Vlasov dispersion theory in a homogeneous electron-proton plasma implies an instability threshold condition at constant maximum growth rate 1-T⊥p/T∥p=Sp/&bgr;∥p&agr;p over 1<&bgr;∥p≤10 where &bgr;∥p≡8&pgr;npT∥p/B02 and B0 is the background magnetic field. Here Sp and &agr;p are fitting parameters and &agr;p≃0.7. One- and two-dimensional initial value hybrid simulations of this growing mode are carried out under proton cyclotron resonant conditions in a homogeneous plasma on the initial domain 2≲&bgr;∥p≤100. The two-dimensional simulations show that enhanced fluctuations from this instability impose a bound on the proton temperature anisotropy of the form of the above equation with the fluid theory result &agr;p≃1.0. On this domain both one- and two-dimensional simulations yield a new form for the upper bound on the fluctuating field energy density from the proton resonant firehose instability |ΔB|2/B02=SB+&agr;B ln(&bgr;∥p) where SB and &agr;B are empirical parameters which are functions of the initial growth rate. This logarithmic behavior is qualitatively different from a fluid theory prediction and, like the anisotropy bound, should be subject to observational verification in any sufficiently homogeneous plasma in which the proton velocity distribution is approximately bi-Maxwellian. ¿ 1998 American Geophysical Union