  We present a finite temperature ($T$) study of the t-J model on the
two-dimensional triangular lattice for the negative hopping $t$, as relevant
for the electron-doped Na$_x$CoO$_2$ (NCO). To understand several aspects of
this system, we study the $T$-dependent chemical potential, specific heat,
magnetic susceptibility, and the dynamic Hall-coefficient across the entire
doping range. We show systematically, how this simplest model for strongly
correlated electrons describes a crossover as function of doping ($x$) from a
Pauli-like weakly spin-correlated metal close to the band-limit (density $n=2$)
to the Curie-Weiss metallic phase ($1.5<n<1.75$) with pronounced
anti-ferromagnetic (AFM) correlations at low temperatures and Curie-Weiss type
behavior in the high-temperature regime. Upon further reduction of the doping,
a new energy scale, dominated by spin-interactions ($J$) emerges (apparent both
in specific heat and susceptibility) and we identify an effective interaction
$J_{eff}(x)$, valid across the entire doping range. This is distinct from
Anderson's formula, as we choose here $t<0$, hence the opposite sign of the
usual Nagaoka-ferromagnetic situation. This expression includes the subtle
effect of weak kinetic AFM - as encountered in the infinitely correlated
situation ($U=\infty$). By explicit computation of the Kubo-formulae, we
address the question of practical relevance of the high-frequency expression
for the Hall coefficient $R_H^*$. We hope to clarify some open questions
concerning the applicability of the t-J model to real experimental situations
through this study.
