In this study, natural convection flow in a porous cavity with sinusoidal temperature distribution has been analyzed by a new double multi relaxation time (MRT) Lattice Boltzmann method (LBM). We consider a copper/water nanofluid filling a porous cavity. For simulating the temperature and flow fields, D2Q5 and D2Q9 lattices are utilized respectively, and the effects of different Darcy numbers (Da) (0.001-0.1) and various Rayleigh numbers (Ra) ($10^3$-$10^5$) for porosity ($ε$) between 0.4 and 0.9 have been considered. Phase deviation ($θ$) changed from 0 to $π$ and the volume fraction of nanoparticles (Ø) varied from 0 to 6%. The present results show a good agreement with the previous works, thus confirming the reliability the new numerical method proposed in this paper. It is indicated that the heat transfer rate increases at increasing Darcy number, porosity, Rayleigh number, the volume fraction of nanoparticles and phase deviation. However, the most sensitive parameter is the Rayleigh number. The maximum Nusselt deviation is 10%, 32% and 33% for Ra=$10^3$, $10^4$ and $10^5$, respectively, with $ε = 0.4$ to $ε = 0.9$. It can be concluded that the effect of Darcy number on the heat transfer rate increases at increasing Rayleigh number, yielding a maximum enhancement of the average Nusselt number around 12% and 61% for Ra=$10^3$ and Ra=$10^5$, respectively.