Modified first law of 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 Psi^A da, where Psi^A is the thermodynamic potential conjugate to a. We derive 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 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.