In this paper we introduce, according to one of the main ideas of $\tau$-tilting theory, the $\tau$-tilting Hochschild cohomology in degree one of a finite dimensional $k$-algebra $\mathrm{\Lambda }$, where $k$ is a field. We define the excess of $\mathrm{\Lambda }$ as the difference between the dimensions of the $\tau$-tilting Hochschild cohomology in degree one and the dimension of the usual Hochschild cohomology in degree one.

One of the main results is that for a zero excess bound quiver algebra $\mathrm{\Lambda }\mathrm{=}k\mathsf{}Q\mathrm{/}I$, the Hochschild cohomology in degree two $H\mathsf{}{H}^{\mathrm{2}}\mathsf{}\mathrm{(}\mathrm{\Lambda }\mathrm{)}$ is isomorphic to the space of morphisms ${\mathrm{𝖧𝗈𝗆}}_{k\mathsf{}Q\mathrm{-}k\mathsf{}Q}\mathsf{}\mathrm{(}I\mathrm{/}{I}^{\mathrm{2}}\mathrm{,}\mathrm{\Lambda }\mathrm{)}\mathrm{.}$ This may be useful to determine when $H\mathsf{}{H}^{\mathrm{2}}\mathsf{}\mathrm{(}\mathrm{\Lambda }\mathrm{)}\mathrm{=}\mathrm{0}$ for these algebras.

We compute the excess for hereditary, radical square zero and monomial triangular algebras. For a bound quiver algebra $\mathrm{\Lambda }$, a formula for the excess of $\mathrm{\Lambda }$ is obtained. We also give a criterion for $\mathrm{\Lambda }$ to be $\tau$-rigid.