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The zeroth law of thermodynamics involves a transitivity relation (pairwisebetween three objects) expressed either in terms of `equal temperature' (ET),or `in equilibrium' (EQ) conditions. In conventional thermodynamics conditionalon vanishingly weak system-bath coupling these two conditions are commonlyregarded as equivalent. In this work we show that for thermodynamics at strongcoupling they are inequivalent: namely, two systems can be in equilibrium andyet have different effective temperatures. A recent result \cite{NEqFE} forGaussian quantum systems shows that an effective temperature $T^{*}$ can bedefined at all times during a system's nonequilibrium evolution, but because ofthe inclusion of interaction energy, after equilibration the system's $T^*$ isslightly higher than the bath temperature $T_{\textsc{b}}$, with the deviationdepending on the coupling. A second object coupled with a different strengthwith an identical bath at temperature $T_{\textsc{b}}$ will not have the sameequilibrated temperature as the first object. Thus $ET \neq EQ $ for strongcoupling thermodynamics. We then investigate the conditions for dynamicalequilibration for two objects 1 and 2 strongly coupled with a common bath $B$,each with a different equilibrated effective temperature. We show this ispossible, and prove the existence of a generalized fluctuation-dissipationrelation under this configuration. This affirms that `in equilibrium' is avalid and perhaps more fundamental notion which the zeroth law for quantumthermodynamics at strong coupling should be based on. Only when the system-bathcoupling becomes vanishingly weak that `temperature' appearing in thermodynamicrelations becomes universally defined and makes better physical sense.

Zeroth Law in Quantum Thermodynamics at Strong Coupling: `in Equilibrium', not `Equal Temperature'