(Abridged) We present a spectral analysis of a deep (220 ks) XMM-Newton observation of the Phoenix cluster (SPT-CL J2344-4243), which we also combine with Chandra archival ACIS-I data. We extract CCD and RGS X-ray spectra from the core region to search for the signature of cold gas, and constrain the mass deposition rate in the cooling flow which is thought to be responsible of the massive star formation episode observed in the BCG. We find an average mass deposition rate of $\dot M = 620 (-190 +200)_{stat} (-50 +150)_{syst} M_⊙$/yr in the temperature range 0.3-3.0 keV from MOS data. A temperature-resolved analysis shows that a significant amount of gas is deposited only above 1.8 keV, while upper limits of the order of hundreds of $M_⊙$/yr can be put in the 0.3-1.8 keV temperature range. From pn data we obtain $\dot M = 210 (-80 +85)_{stat} ( -35 +60)_{syst} M_⊙$/yr, and the upper limits from the temperature-resolved analysis are typically a factor of 3 lower than MOS data. In the RGS spectrum, no line emission from ionization states below Fe XXIII is seen above $12 Å$, and the amount of gas cooling below $∼ 3$ keV has a best-fit value $\dot M = 122_{-122}^{+343}$ $M_{⊙}$/yr. In addition, our analysis of the FIR SED of the BCG based on Herschel data provides $SFR = (530 ± 50) M_⊙$/yr, significantly lower than previous estimates by a factor 1.5. Current data are able to firmly identify substantial amount of cooling gas only above 1.8 keV in the core of the Phoenix cluster. While MOS data analysis is consistent with values as high as $\dot M ∼ 1000$ within $1 σ$, pn data provide $\dot M