We propose a phenomenological modification to the classical Bekenstein–Hawking entropy-area relation by introducing a square-root correction inspired by quantum gravitational considerations. The modified entropy, of the form \( S = \frac{A}{4} \left(1 + \lambda \sqrt{A} \right) \), leads to significant changes in the thermodynamics of Schwarzschild black holes, including a suppressed Hawking temperature, a less negative specific heat, and a decelerated evaporation rate. These effects indicate the existence of stable black hole remnants, consistent with predictions from generalized uncertainty principles and noncommutative geometry. We discuss the implications for the black hole information paradox and the potential relevance of such remnants to dark matter phenomenology. The model offers a novel yet minimal extension to classical thermodynamics, bridging semiclassical and quantum gravity regimes.