Using the linear response formalism for the spin-orbit torque (SOT) we compute from first principles the SOT in a system of two layers of L1$_0$-FePt(001) deposited on an fcc Pt(001) substrate of varying thickness. We find that at room temperature the values of the SOTs that are even and odd with respect to magnetization generally lie in the range of values measured and computed for Co/Pt bilayers. We also observe that the even SOT is much more robust with respect to changing the number of layers in the substrate, and as a function of energy it follows the general trend of the even SOT exerted by the spin Hall current in fcc Pt. The odd torque, on the other hand, is strongly affected by modification of the electronic structure for a specific energy window in the limit of very thin films. Moreover, taking the system at hand as an example, we compute the values of the thermal spin-orbit torque (T-SOT). We predict that the gradients of temperature which can be experimentally created in this type of systems will cause a detectable torque on the magnetization. We also underline the correlation between the even T-SOT and the spin Nernst effect, thus motivating a more intensive experimental effort aimed at observation of both phenomena. ; Comment: 8 pages, 4 figures