  We study the statistical physics of the classical Ising model in the
so-called $\alpha$-R\'enyi ensemble, a finite-temperature thermal state
approximation that minimizes a modified free energy based on the
$\alpha$-R\'enyi entropy. We begin by characterizing its critical behavior in
mean-field theory in different regimes of the R\'enyi index $\alpha$. Next, we
re-introduce correlations and consider the model in one and two dimensions,
presenting an exact analysis of the former and devising an unconventional Monte
Carlo approach to the study of the latter. Remarkably, we find that while
mean-field predicts a continuous phase transition below a threshold index value
of $\alpha \sim 1.303$ and a first-order transition above it, the Monte Carlo
results in two dimensions point to a continuous transition at all $\alpha$. We
conclude by performing a variational minimization of the $\alpha$-R\'enyi free
energy using a recurrent neural network (RNN) ansatz where we find that the RNN
performs well in two dimensions when compared to the Monte Carlo simulations.
Our work highlights the potential opportunities and limitations associated with
the use of the $\alpha$-R\'enyi ensemble formalism in probing the thermodynamic
equilibrium properties of classical and quantum systems.
