  Motivated by the recently observed unconventional Hall effect in ultra-thin
films of ferromagnetic SrRuO$_3$ (SRO) we investigate the effect of
strain-induced oxygen octahedral distortion in the electronic structure and
anomalous Hall response of the SRO ultra-thin films by virtue of density
functional theory calculations. Our findings reveal that the ferromagnetic SRO
films grown on SrTiO$_3$ (in-plane strain of $-$0.47$\%$) have an orthorhombic
(both tilting and rotation) distorted structure and with an increasing amount
of substrate-induced compressive strain the octahedral tilting angle is found
to be suppressed gradually, with SRO films grown on NdGaO$_3$ (in-plane strain
of $-$1.7$\%$) stabilized in the tetragonal distorted structure (with zero
tilting). Our Berry curvature calculations predict a positive value of the
anomalous Hall conductivity of $+$76\,S/cm at $-$1.7$\%$ strain, whereas it is
found to be negative ($-$156\,S/cm) at $-$0.47$\%$ strain. We attribute the
found behavior of the anomalous Hall effect to the nodal point dynamics in the
electronic structure arising in response to tailoring the oxygen octahedral
distortion driven by the substrate-induced strain. We also calculate
strain-mediated anomalous Hall conductivity as a function of reduced
magnetization obtained by scaling down the magnitude of the exchange field
inside Ru atoms finding good qualitative agreement with experimental
observations, which indicates a strong impact of longitudinal thermal
fluctuations of Ru spin moments on the anomalous Hall effect in this system.
