  The tunneling approach beyond semiclassical approximation has been used to
calculate the corrected Hawking temperature and entropy for various black holes
and FRW universe model. We examine their derivations, and prove that the
quantity $H$ in the corrected temperature is the explicit function of the only
free parameter $\mathcal{A}$ (which is an auxiliary parameter defined by
$\mathcal{A}=\hbar S_{BH}$). Our analysis improves previous calculations, and
indicates that the leading order logarithmic correction to entropy is a natural
result of the corrected temperature and the first law of thermodynamics.
Additionally, we apply the tunneling approach beyond semiclassical
approximation to neutral black rings. Based on the analysis, we show that the
entropy of neutral black rings also has a logarithmic leading order correction.
