It was recently argued that $f(T)$ gravity could inherit “remnant symmetry” from the full Lorentz group, despite the fact that the theory is not locally Lorentz invariant. Confusion has arisen regarding the implication of this result for the previous works, which established that $f(T)$ gravity is pathological due to superluminal propagation, local acausality, and nonunique time evolution. We clarify that the existence of the “remnant group” does not rid the theory of these various problems but, instead, strongly supports it.

Barrow proposed that the area law of the entropy associated with a horizon might receive a “fractal correction” due to quantum gravitational effects—in place of $S\propto A$, we have instead $S\propto A^{1+\delta /2}$, where $0⩽\delta ⩽1$ measures the deviation from the standard area law ($\delta =0$). Based on black hole thermodynamics, we argue that the Barrow entropy should run (i.e., energy scale dependent), which is reasonable given that quantum gravitational corrections are expected to be important only in the high-energy regime. When applied to the Friedmann equation, we demonstrate the possibility that such a running Barrow entropy index could give rise to a dynamical effective dark energy, which is asymptotically positive and vanishing, but negative at the Big Bang. Such a sign switching dark energy could help to alleviate the Hubble tension. Other cosmological implications are discussed.