Here, we explore the role of inherent altermagnetic topology in transverse transport phenomena (such as crystal or anomalous Hall, Nernst, and thermal Hall effects) in several famous altermagnets, including tetragonal $X{\mathrm{V}}_{2}Y{}_2\mathrm{O}$ ($X=\mathrm{K},\mathrm{Rb},\mathrm{Cs}$; $Y=\mathrm{S},\mathrm{Se},\mathrm{Te}$), ${\mathrm{RuO}}_2, {\mathrm{MnF}}_2$, as well as hexagonal CrSb and MnTe. Notably, in $X{\mathrm{V}}_{2}Y{}_2\mathrm{O}$, the first experimentally realized layered altermagnets, transverse transport is governed by altermagnetic pseudonodal surfaces, emphasizing the purely topological contributions to transverse transport. Interestingly, we demonstrate that strain engineering and magnetic field, two unique methods for selectively controlling crystal and anomalous transport, can substantially enhance the magnitude of these phenomena while preserving the alternating spin characteristics in both real and momentum space. Moreover, due to the spin symmetry breaking via shear strain, a new magnetic phase, fully compensated ferrimagnetism, with isotropic spin splitting, can be induced. Our findings provide effective strategies not only for manipulating transverse transport in altermagnets but also for controlling magnetic phase transitions, offering valuable insights for their potential applications in spintronics and spin caloritronics.

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• Anomalous Hall effect
• Nernst effect
• Thermal Hall effect
• Altermagnets
• Topological materials