  We consider the compactification on a circle of the Heterotic Superstring
effective action to first order in the Regge slope parameter $\alpha'$ and
re-derive the $\alpha'$-corrected Buscher rules first found in
arXiv:hep-th/9506156 , proving the T duality invariance of the
dimensionally-reduced action to that order in $\alpha'$. We use Iyer and Wald's
prescription to derive an entropy formula that can be applied to black-hole
solutions which can be obtained by a single non-trivial compactification on a
circle and discuss its invariance under the $\alpha'$-corrected T duality
transformations. This formula has been successfully applied to
$\alpha'$-corrected 4-dimensional non-extremal Reissner-Nordstr\"om black holes
in arXiv:1910.14324 and we apply it here to a heterotic version of the
Strominger-Vafa 5-dimensional extremal black hole.
