  The zeroth law of thermodynamics involves a transitivity relation (pairwise
between three objects) expressed either in terms of `equal temperature' (ET),
or `in equilibrium' (EQ) conditions. In conventional thermodynamics conditional
on vanishingly weak system-bath coupling these two conditions are commonly
regarded as equivalent. In this work we show that for thermodynamics at strong
coupling they are inequivalent: namely, two systems can be in equilibrium and
yet have different effective temperatures. A recent result \cite{NEqFE} for
Gaussian quantum systems shows that an effective temperature $T^{*}$ can be
defined at all times during a system's nonequilibrium evolution, but because of
the inclusion of interaction energy, after equilibration the system's $T^*$ is
slightly higher than the bath temperature $T_{\textsc{b}}$, with the deviation
depending on the coupling. A second object coupled with a different strength
with an identical bath at temperature $T_{\textsc{b}}$ will not have the same
equilibrated temperature as the first object. Thus $ET \neq EQ $ for strong
coupling thermodynamics. We then investigate the conditions for dynamical
equilibration for two objects 1 and 2 strongly coupled with a common bath $B$,
each with a different equilibrated effective temperature. We show this is
possible, and prove the existence of a generalized fluctuation-dissipation
relation under this configuration. This affirms that `in equilibrium' is a
valid and perhaps more fundamental notion which the zeroth law for quantum
thermodynamics at strong coupling should be based on. Only when the system-bath
coupling becomes vanishingly weak that `temperature' appearing in thermodynamic
relations becomes universally defined and makes better physical sense.
