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Absence of isolated critical points with nonstandard critical exponents in the four-dimensional regularization of Lovelock gravity

Hyperbolic vacuum black holes in Lovelock gravity theories of odd order $N$, in which $N$ denotes the order of higher-curvature corrections, are known to have the so-called isolated critical points with nonstandard critical exponents (as $α= 0$, $β= 1$, $γ= N-1$, and $δ= N$), different from those of mean-field critical exponents (with $α= 0$, $β= 1/2$, $γ= 1$, and $δ= 3$). Motivated by this important observation, here, we explore the consequences of taking the $D \to 4$ limit of Lovelock gravity and the possibility of finding nonstandard critical exponents associated with isolated critical points in four-dimensions by use of the four-dimensional regularization technique, proposed recently by Glavan and Lin \cite{Glavan2020}. To do so, we first present $\text{AdS}_4$ Einstein-Lovelock black holes with fine-tuned Lovelock couplings in the regularized theory, which is needed for our purpose. Next, it is shown that the regularized $4D$ Einstein-Lovelock gravity theories of odd order $N > 3$ do not possess any physical isolated critical point, unlike the conventional Lovelock gravity. In fact, the critical (inflection) points of the equation of state always occur for the branch of black holes with negative entropy. The situation is quite different for the case of the regularized $4D$ Einstein-Lovelock gravity with cubic curvature corrections ($N=3$). In this case ($N=3$), although the entropy is non-negative and the equation of state of hyperbolic vacuum black holes has a nonstandard Taylor expansion about its inflection point, but there is no criticality associated with this special point. At the inflection point, the physical properties of the black hole system change drastically ...