We report here on the outburst onset and evolution of the new Soft Gamma Repeater SGR 0501+4516 . We monitored the new SGR with XMM–Newton starting on 2008 August 23 , one day after the source became burst-active , and continuing with 4 more observations in the following month , with the last one on 2008 September 30 . Combining the data with the Swift -XRT and Suzaku data we modelled the outburst decay over a three months period , and we found that the source flux decreased exponentially with a timescale of t _ { c } = 23.8 days . In the first XMM–Newton observation a large number of short X-ray bursts were observed , the rate of which decayed drastically in the following observations . We found large changes in the spectral and timing behavior of the source during the first month of the outburst decay , with softening emission as the flux decayed , and the non-thermal soft X-ray spectral component fading faster than the thermal one . Almost simultaneously to our second and fourth XMM–Newton observations ( on 2008 August 29 and September 2 ) , we observed the source in the hard X-ray range with INTEGRAL , which clearly detected the source up to \sim 100keV in the first pointing , while giving only upper limits during the second pointing , discovering a variable hard X-ray component fading in less than 10 days after the bursting activation . We performed a phase-coherent X-ray timing analysis over about 160 days starting with the burst activation and found evidence of a strong second derivative period component ( \ddot { P } = -1.6 ( 4 ) \times 10 ^ { -19 } s s ^ { -2 } ) . Thanks to the phase-connection , we were able to study the the phase-resolved spectral evolution of SGR 0501+4516 in great detail . We also report on the \R quiescent source data , taken back in 1992 when the source exhibits a flux \sim 80 times lower than that measured during the outburst , and a rather soft , thermal spectrum .