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We give a new and conceptually simple approach to obtain the first law ofblack hole thermodynamics from a basic thermodynamical property that entropy(S) for any stationary black hole is a state function implying that dS must bean exact differential. Using this property we obtain some conditions which areanalogous to Maxwell's relations in ordinary thermodynamics. From theseconditions we are able to explicitly calculate the semiclassicalBekenstein-Hawking entropy, considering the most general metric represented bythe Kerr-Newman spacetime. We extend our method to find the corrected entropyof stationary black holes in (3+1) dimensions. For that we first calculate thecorrected Hawking temperature considering both scalar particle and fermiontunneling beyond the semiclassical approximation. Using this corrected Hawkingtemperature we compute the corrected entropy, based on properties of exactdifferentials. The connection of the coefficient of the leading (logarithmic)correction with the trace anomaly of the stress tensor is established . Weexplicitly calculate this coefficient for stationary black holes with variousmetrics, emphasising the role of Komar integrals.

Exact Differential and Corrected Area Law for Stationary Black Holes in Tunneling Method