The anomalous thermodynamic properties of the paradigmatic frustrated spin-1/2 triangular lattice Heisenberg antiferromagnet (TLH) has remained an open topic of research over decades, both experimentally and theoretically. Here we further the theoretical understanding based on the recently developed, powerful exponential tensor renormalization group (XTRG) method on cylinders and stripes in a quasi one-dimensional (1D) setup, as well as a tensor product operator approach directly in 2D. The observed thermal properties of the TLH are in excellent agreement with two recent experimental measurements on the virtually ideal TLH material Ba$_8$CoNb$_6$O$_{24}$. Remarkably, our numerical simulations reveal two crossover temperature scales, at $T_l/J sim 0.20$ and $T_h/Jsim 0.55$, with $J$ the Heisenberg exchange coupling, which are also confirmed by a more careful inspection of the experimental data. We propose that in the intermediate regime between the low-temperature scale $T_l$ and the higher one $T_h$, the gapped roton-like excitations are activated with a strong chiral component and a large contribution to thermal entropies, which suppress the incipient 120$^circ$ order that emerges for temperatures below $T_l$.
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