Modified first law of a charged dilaton black hole

We investigate the thermodynamics of a charged dilaton black hole arising from Einstein--Maxwell--dilaton theory, where the dilaton couples exponentially to the Maxwell field via a dimensionless parameter $a$. Treating $a$ as a continuous solution parameter, we extend the black hole first law to include a term ${\mathrm{\ensuremath{\Psi}}}^{A}da$, where ${\mathrm{\ensuremath{\Psi}}}^{A}$ is the thermodynamic potential conjugate to $a$. We derive ${\mathrm{\ensuremath{\Psi}}}^{A}$ explicitly through a differential analysis of the mass, charge, and entropy and confirm its form via an independent Hamiltonian calculation. Additionally, by promoting $a$ to a spacetime-dependent scalar and introducing auxiliary gauge fields, we provide a geometric interpretation of ${\mathrm{\ensuremath{\Psi}}}^{A}$ as a conserved Noether charge. We further analyze the implications of treating $a$ as a thermodynamic variable within the extended phase space. Despite the modification to the first law, we demonstrate that the Smarr relation remains unaffected due to the dimensionless nature of $a$, highlighting the distinction between variational and scaling symmetries. Our analysis supports the thermodynamic relevance of coupling constants and enriches the framework of extended black hole thermodynamics.