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In this study, we investigate the position and momentum Shannon entropy, denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula>, respectively, in the context of the fractional Schrödinger equation (FSE) for a hyperbolic double well potential (HDWP). We explore various values of the fractional derivative represented by <i>k</i> in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and the momentum entropy density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for low-lying states. Specifically, as the fractional derivative <i>k</i> decreases, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> becomes more localized, whereas <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> becomes more delocalized. Moreover, we observe that as the derivative <i>k</i> decreases, the position entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> decreases, while the momentum entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative <i>k</i>. It is noteworthy that, despite the increase in position Shannon entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> and the decrease in momentum Shannon entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> with an increase in the depth <i>u</i> of the HDWP, the Beckner–Bialynicki-Birula–Mycielski (BBM) inequality relation remains satisfied. Furthermore, we examine the Fisher entropy and its dependence on the depth <i>u</i> of the HDWP and the fractional derivative <i>k</i>. Our results indicate that the Fisher entropy increases as the depth <i>u</i> of the HDWP is increased and the fractional derivative <i>k</i> is decreased.