The presence of multi-component protons with their distinct features is confirmed by various space missions in the Earth’s outer magnetosphere regions. Isotropic cold protons and anisotropic hot protons significantly influence/modify the dispersion behaviour of various modes and instabilities and regulate the magnetospheric dynamics effectively. Our present study pays attention to the left-hand-polarised proton cyclotron mode, which gets unstable in the large proton temperature anisotropy condition, i.e.
$T_{\perp p}\gt T_{\parallel p}$. Such favourable thermal conditions for protons are extensively observed during the compression of the solar wind against the Earth’s magnetic field. To reveal the wave dynamics in more detail, i.e. time-scale variations in the cold and hot proton temperatures and resulting wave-energy density, we further allow the time evolution of our model bi-Maxwellian distribution function in response to the proton cyclotron instability. Based on velocity-moment techniques, we formulated a set of equations comprising an instantaneous dispersion relation, dynamical perpendicular and parallel temperature relations and a wave-energy density equation. For the graphical illustrations of our mathematical results, we choose initial conditions that are relevant to magnetospheric space environments and reported in various experimental studies. Our exact numerical analysis shows the notable impact of hot proton temperature anisotropy and relative density on the real frequency, growth rate, evolution of initial distributions and wave-energy density of the proton cyclotron instability. Such detailed outcomes will be quite helpful for global/local magnetospheric experimental and simulation studies.