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It is shown numerically that for Fermi Pasta Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N^(1/3), relevant for the much debated question of thedoi:10.1103/physrevlett.98.184301 pmid:17501580 fatcat:x64eqfm2ijbb7dmdgewuc2tktm