A simple example is obtained by taking $P$ to mean "has positive dimension".
Every local domain of positive dimension $(A,\mathfrak m)$ has $P$ at all maximal ideals (i.e. just at $\mathfrak m$ !) since $A_{\mathfrak m}=A$ , but $P$ fails at the generic point $\eta=(0)$ since $A_\eta=Frac(A)$ has dimension zero, being a field.

Edit In order to address Chuck's comment, let me emphasize that the answer above is very easily adapted to non local non-local rings.
For example any finitely generated domain $A$ of positive dimension $d$ over a field has property $P$ when localized at a maximal ideal $\frak m$ but not at the zero ideal.
More precisely, $dim A_{(0)}=0$ and $dimA_{\frak m}=d$ for any maximal ideal ${\frak m}$ : this equidimensionality result follows from Noether's normalization theorem.
This shows that if property $P_d$ is " has dimension $d$ ", then $P_d$ holds for $A_{\frak m}$ if ${\frak m}$ is maximal and does not hold for $A_{\frak p}$ if the prime $\frak p$ is not maximal.

A simple example is obtained by taking $P$ to mean "has positive dimension".
Every local domain of positive dimension $(A,\mathfrak m)$ has $P$ at all maximal ideals (i.e. just at $\mathfrak m$ !) since $A_{\mathfrak m}=A$ , but $P$ fails at the generic point $\eta=(0)$ since $A_\eta=Frac(A)$ has dimension zero, being a field.

Edit In order to address Chuck's comment, let me emphasize that the answer above is very easily adapted to non local rings.
For example any finitely generated domain $A$ of positive dimension $d$ over a field has property $P$ when localized at a maximal ideal $\frak m$ but not at the zero ideal.
More precisely, $dim A_{(0)}=0$ and $dimA_{\frak m}=d$ for any maximal ideal ${\frak m}$ : this equidimensionality result follows from Noether's normalization theorem.

3 added 8 characters in body

A simple example is obtained by taking $P$ to mean "has positive dimension".
Every local domain of positive dimension $(A,\mathfrak m)$ has $P$ at all maximal ideals (i.e. just at $\mathfrak m$ !) since $A_{\mathfrak m}=A$ , but $P$ fails at the generic point $\eta=(0)$ since $A_\eta$ A_\eta=Frac(A)\$ has dimension zero, being a field.

2 Replaced "integral ring" by "domain"
1